From ba3c620112a603f6384c30e34419161c6e0e6d18 Mon Sep 17 00:00:00 2001 From: Kyle Niemeyer Date: Thu, 27 Feb 2020 00:00:13 -0800 Subject: [PATCH] more eigenvalue content --- _build/bvps/eigenvalue.html | 457 +++++++++++++++++++- _build/bvps/finite-difference.html | 9 +- _build/bvps/masses.m | 1 + _build/bvps/shooting-method.html | 22 +- _build/images/bvps/eigenvalue_14_0.png | Bin 13846 -> 19563 bytes _build/images/bvps/eigenvalue_16_0.png | Bin 0 -> 9959 bytes _build/images/bvps/shooting-method_13_0.png | Bin 12576 -> 12576 bytes _build/images/bvps/shooting-method_3_1.png | Bin 8763 -> 8763 bytes content/bvps/eigenvalue.ipynb | 342 ++++++++++++++- content/bvps/finite-difference.ipynb | 14 +- content/bvps/masses.m | 1 + content/bvps/shooting-method.ipynb | 23 +- 12 files changed, 804 insertions(+), 65 deletions(-) create mode 100644 _build/images/bvps/eigenvalue_16_0.png diff --git a/_build/bvps/eigenvalue.html b/_build/bvps/eigenvalue.html index 71cea06..0aa90ff 100644 --- a/_build/bvps/eigenvalue.html +++ b/_build/bvps/eigenvalue.html @@ -11,7 +11,7 @@ next_page: url: /quizzes/quiz2-IVPs.html suffix: .ipynb -search: y lambda begin align end x l k frac equation n pi eigenvalues p beam delta b boundary quad rightarrow bmatrix solution conditions ei cos sin lets case our values get ldots infty left right different equations mathbf det eigenvalue problems where buckling consider deflection mz modes gather load yi obtain us system example supported also e ode general neq because instead need associated represent corresponding cr finite above using matrix given means characteristic value not analytical certain simply governing considering sum simplify trivial text problem infinite eigenfunctions three buckle recall properties sqrt critical slightly same differences points into modify +search: y x align begin end lambda k frac l equation n omega m pi eigenvalues left right bmatrix system p b delta beam sin quad lets conditions boundary equations rightarrow different modes det t solution values ei cos case our mathbf get prime example ldots infty mode above into eigenvalue problems where buckling consider deflection mz also need associated represent gather load using yi mass motion masses amplitude odes initial obtain us supported e d ode general neq because instead corresponding connected cr same based finite matrix calculate given spring amplitudes means characteristic value not analytical certain simply governing considering comment: "***PROGRAMMATICALLY GENERATED, DO NOT EDIT. SEE ORIGINAL FILES IN /content***" --- @@ -74,9 +74,9 @@

Example: beam buckling
-
clear all; clc
+
clear all; clc
 
-L = 1.0;
+L = 1.0;
 x = linspace(0, L);
 subplot(1,3,1);
 y = sin(pi * x / L);
@@ -245,18 +245,18 @@ 
Getting eigenvalues numerically
     

-
clear all; clc
+
clear all; clc
 
-dx = 1.0;
+dx = 1.0;
 L = 3.0;
 
 Astar = [2 -1; -1 2];
 k = eig(Astar);
 
-lambda = sqrt(k) / dx^2;
+lambda = sqrt(k) / dx;
 
-fprintf('lambda 1: %6.3f\n', lambda(1));
-fprintf('lambda 2: %6.3f', lambda(2));
+fprintf('lambda_1: %6.3f\n', lambda(1));
+fprintf('lambda_2: %6.3f', lambda(2));
 

 
     

@@ -270,8 +270,8 @@ 
Getting eigenvalues numerically
 
 

-
lambda 1:  1.000
-lambda 2:  1.732

+
lambda_1:  1.000
+lambda_2:  1.732

 

 

 

@@ -281,6 +281,443 @@ 
Getting eigenvalues numerically
+
+ +
+
+

As expected, this matches with our manual calculation above. But, we might want to calculate these eigenvalues more accurately, so let's generalize this a bit and then try using $\Delta x= 0.1$:

+ +
+
+
+
+ +
+ +
+
+ +
+
+
clear all; clc
+
+dx = 0.1;
+L = 3.0;
+x = 0 : dx : L;
+n = length(x) - 2;
+
+Astar = zeros(n,n);
+for i = 1 : n
+    if i == 1
+        Astar(1,1) = 2;
+        Astar(1,2) = -1;
+    elseif i == n
+        Astar(n,n-1) = -1;
+        Astar(n,n) = 2;
+    else
+        Astar(i,i-1) = -1;
+        Astar(i,i) = 2;
+        Astar(i,i+1) = -1;
+    end
+end
+k = eig(Astar);
+
+lambda = sqrt(k) / dx;
+
+fprintf('lambda_1: %6.3f\n', lambda(1));
+fprintf('lambda_2: %6.3f\n\n', lambda(2));
+
+err = abs(lambda(1) - (pi/L)) / (pi/L);
+fprintf('Error in lambda_1: %5.2f%%\n', 100*err);
+
+ +
+
+
+ +
+
+ +
+
+ +
+
lambda_1:  1.047
+lambda_2:  2.091
+
+Error in lambda_1:  0.05%
+
+
+
+
+
+
+ +
+
+ +
+ +
+
+

Example: mass-spring system

Let's analyze the motion of masses connected by springs in a system:

+

+
+ mass-spring system +
Figure: System with two masses connected by springs
+
+
+First, we need to write the equations of motion, based on doing a free-body diagram on each mass: +\begin{align} +m_1 \frac{d^2 x_1}{dt^2} &= -k x_1 + k(x_2 - x_1) \\ +m_2 \frac{d^2 x_2}{dt^2} &= -k (x_2 - x_1) - k x_2 +\end{align} +We can condense these equations a bit: +\begin{align} +x_1^{\prime\prime} - \frac{k}{m_1} \left( -2 x_1 + x_2 \right) &= 0 \\ +x_2^{\prime\prime} - \frac{k}{m_2} \left( x_1 - 2 x_2 \right) &= 0 +\end{align}

+

To proceed, we can assume that the masses will move in a sinusoidal fashion, with a shared frequency but separate amplitude: +\begin{align} +x_i &= A_i \sin (\omega t) \\ +x_i^{\prime\prime} &= -A_i \omega^2 \sin (\omega t) +\end{align} +We can plug these into the ODEs: +\begin{align} +\sin (\omega t) \left[ \left( \frac{2k}{m_1} - \omega^2 \right) A_1 - \frac{k}{m_1} A_2 \right] &= 0 \\ +\sin (\omega t) \left[ -\frac{k}{m_2} A_1 + \left( \frac{2k}{m_2} - \omega^2 \right) A_2 \right] &= 0 +\end{align} +or +\begin{align} +\left( \frac{2k}{m_1} - \omega^2 \right) A_1 - \frac{k}{m_1} A_2 &= 0 \\ +-\frac{k}{m_2} A_1 + \left( \frac{2k}{m_2} - \omega^2 \right) A_2 &= 0 +\end{align} +Let's put some numbers in, and try to solve for the eigenvalues: $\omega^2$. +Let $m_1 = m_2 = 40 $ kg and $k = 200$ N/m.

+

Now, the equations become +\begin{align} +\left( 10 - \omega^2 \right) A_1 - 5 A_2 &= 0 \\ +-5 A_1 + \left( 10 - \omega^2 \right) A_2 &= 0 +\end{align} +or $A \mathbf{y} = \mathbf{0}$, which we can represent as +\begin{equation} +\begin{bmatrix} 10-\omega^2 & -5 \\ -5 & 10-\omega^2 \end{bmatrix} +\begin{bmatrix} A_1 \\ A_2 \end{bmatrix} = +\begin{bmatrix} 0 \\ 0 \end{bmatrix} +\end{equation} +Here, $\omega^2$ are the eigenvalues, and we can find them with $\det(A) = 0$: +\begin{align} +\det(B) &= 0 \\ +\det (B^* - \omega^2 I) &= 0 +\end{align}

+ +
+
+
+
+ +
+ +
+
+ +
+
+
clear all; clc
+
+Bstar = [10 -5; -5 10];
+omega_squared = eig(Bstar);
+omega = sqrt(omega_squared);
+
+fprintf('omega_1 = %5.2f rad/s\n', omega(1));
+fprintf('omega_2 = %5.2f rad/s\n', omega(2));
+
+ +
+
+
+ +
+
+ +
+
+ +
+
omega_1 =  2.24 rad/s
+omega_2 =  3.87 rad/s
+
+
+
+
+
+
+ +
+
+ +
+ +
+
+

We find there are two modes of oscillation, each associated with a different natural frequency. Unfortunately, we cannot calculate independent and unique values for the amplitudes, but if we insert the values of $\omega$ into the above equations, we can find relations connecting the amplitudes: +\begin{align} +\omega_1: \quad A_1 &= A_2 \\ +\omega_2: \quad A_1 &= -A_2 +\end{align}

+

So, for the first mode, we have the two masses moving in sync with the same amplitude. In the second mode, they move with opposite (but equal) amplitude. With the two different frequencies, they also have two different periods:

+ +
+
+
+
+ +
+ +
+
+ +
+
+
t = linspace(0, 3);
+subplot(1,5,1)
+plot(sin(omega(1)*t), t); hold on
+plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+box off; set(gca,'Visible','off')
+
+subplot(1,5,2)
+plot(sin(omega(1)*t), t); hold on
+plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+text(-2.5,-0.2, 'First mode')
+box off; set(gca,'Visible','off')
+
+subplot(1,5,4)
+plot(-sin(omega(2)*t), t); hold on
+plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+box off; set(gca,'Visible','off')
+
+subplot(1,5,5)
+plot(sin(omega(2)*t), t); hold on
+plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+box off; set(gca,'Visible','off')
+text(-2.7,-0.2, 'Second mode')
+
+ +
+
+
+ +
+
+ +
+
+ + + +
+ +
+ +
+
+
+
+ +
+
+ +
+ +
+
+

We can confirm that the system would actually behave in this way by setting up the system of ODEs and integrating based on initial conditions matching the amplitudes of the two modes.

+

For example, let's use $x_1 (t=0) = x_2(t=0) = 1$ for the first mode, and $x_1(t=0) = 1$ and $x_2(t=0) = -1$ for the second mode. We'll use zero initial velocity for both cases.

+

Then, we can solve by converting the system of two 2nd-order ODEs into a system of four 1st-order ODEs:

+ +
+
+
+
+ +
+ +
+
+ +
+
+
%%file masses.m
+function dxdt = masses(t, x)
+% this is a function file to calculate the derivatives associated with the system
+
+m1 = 40;
+m2 = 40;
+k = 200;
+
+dxdt = zeros(4,1);
+
+dxdt(1) = x(2);
+dxdt(2) = (k/m1)*(-2*x(1) + x(3));
+dxdt(3) = x(4);
+dxdt(4) = (k/m2)*(x(1) - 2*x(3));
+
+ +
+
+
+ +
+
+ +
+
+ +
+
Created file '/Users/niemeyek/projects/ME373-book/content/bvps/masses.m'.
+
+
+
+
+
+
+ +
+
+ +
+ +
+
+ +
+
+
clear all; clc
+
+% this is the integration for the system in the first mode
+[t, X] = ode45('masses', [0 3], [1.0 0.0 1.0 0.0]);
+subplot(1,5,1)
+plot(X(:,1), t); 
+ylabel('displacement (m)'); xlabel('time (s)')
+set (gca, 'ydir', 'reverse' )
+%box off; set(gca,'Visible','off')
+
+subplot(1,5,2)
+plot(X(:,3), t); xlabel('time (s)')
+set (gca, 'ydir', 'reverse' )
+text(-4,-0.2, 'First mode')
+
+% this is the integration for the system in the second mode
+[t, X] = ode45('masses', [0 3], [1.0 0.0 -1.0 0.0]);
+subplot(1,5,4)
+plot(X(:,1), t);
+ylabel('displacement (m)'); xlabel('time (s)')
+set (gca, 'ydir', 'reverse' )
+%box off; set(gca,'Visible','off')
+
+subplot(1,5,5)
+plot(X(:,3), t); xlabel('time (s)')
+set (gca, 'ydir', 'reverse' )
+text(-4,-0.2, 'Second mode')
+
+ +
+
+
+ +
+
+ +
+
+ + + +
+ +
+ +
+
+
+
+ +
+
+ +
+ +
+
+

This shows that we get either of the pure modes of motion with the appropriate initial conditions.

+

What about if the initial conditions don't match either set of amplitude patterns?

+ +
+
+
+
+ +
+ +
+
+ +
+
+
[t, X] = ode45('masses', [0 3], [0.25 0.0 0.75 0.0]);
+subplot(1,5,1)
+plot(X(:,1), t);
+%plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+%box off; set(gca,'Visible','off')
+
+subplot(1,5,2)
+plot(X(:,3), t);
+set (gca, 'ydir', 'reverse' )
+
+ +
+
+
+ +
+
+ +
+
+ + + +
+ +
+ +
+
+
+
+ +
+
+ +
+ +
+
+

In this case, the resulting motion will be a complicated superposition of the two modes.

+ +
+
+
+
+ diff --git a/_build/bvps/finite-difference.html b/_build/bvps/finite-difference.html index 1bdeb27..5625021 100644 --- a/_build/bvps/finite-difference.html +++ b/_build/bvps/finite-difference.html @@ -11,7 +11,7 @@ next_page: url: /bvps/eigenvalue.html suffix: .ipynb -search: x y delta f t prime equation frac begin end right align boundary left b derivative dx infty m difference xi fin solve equations yi l finite c conditions where solution n heat transfer lets our ac point system condition nonlinear text k p ti differences mathcal o example ode points linear matlab h exact using order second consider gives domain through formula term q dt theta well rightarrow approx into above recursion get bmatrix hand guess temperature d value also accurate five set mathbf implement fixed octave because e control volume method approximations derivatives function approximate forward backward taylor +search: x y delta f prime equation t frac begin end align right boundary left b derivative dx difference xi fin solve equations yi l infty finite c conditions where solution n heat transfer m lets our ac point system condition text k p differences mathcal o example ode points linear matlab nonlinear h exact using order second consider gives domain through formula term q dt theta well rightarrow approx into above recursion get bmatrix guess temperature d value also accurate five set mathbf implement hand fixed octave because e control volume method approximations derivatives function approximate forward backward taylor series comment: "***PROGRAMMATICALLY GENERATED, DO NOT EDIT. SEE ORIGINAL FILES IN /content***" --- @@ -699,12 +699,7 @@

Heat transfer with radiationExample: linear ODE
-
clear all; clc
+
clear all; clc
 
-% target boundary condition
+% target boundary condition
 target = 8;
 
 % Pick a guess for y'(0) of 1
@@ -234,7 +234,7 @@ 
Example: nonlinear ODE
     

 
%%python
-import sympy as sym
+import sympy as sym
 sym.init_printing()
 x, y, u, v = sym.symbols('x y u v')
 
@@ -364,9 +364,9 @@ 
Example: nonlinear ODE
     

-
clear all; clc
+
clear all; clc
 
-target = 1.0;
+target = 1.0;
 
 guesses = zeros(3,1);
 solutions = zeros(3,1);
@@ -443,9 +443,9 @@ 
Example: nonlinear ODE
     

-
clear all; clc
+
clear all; clc
 
-target = 1.0;
+target = 1.0;
 
 % get these arrays of stored values started.
 % note: I'm only doing this to make it easier to show a table of values
@@ -474,9 +474,9 @@ 
Example: nonlinear ODE% we should probably set a maximum number of iterations, just to prevent
     % an infinite while loop in case something goes wrong
-    if num >= 1e4
-        break
-    end
+    if num >= 1e4
+        break
+    end
 end
 
 table(tries, guesses, solutions)
@@ -529,7 +529,7 @@ 
Example: nonlinear ODE
 
%plot -r 200
 plot(F(:, 2), eta); ylim([0 5])
-xlabel("f^{\prime}(\eta) = u/U_{\infty}")
+xlabel("f^{\prime}(\eta) = u/U_{\infty}")
 ylabel('\eta')
 

 
diff --git a/_build/images/bvps/eigenvalue_14_0.png b/_build/images/bvps/eigenvalue_14_0.png
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diff --git a/_build/images/bvps/shooting-method_3_1.png b/_build/images/bvps/shooting-method_3_1.png
index 92ecdd9d90760812e08a44265b9d599f0da72822..affd82e0d043a757f124401c0ce2ede0687577a1 100644
GIT binary patch
delta 47
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DPJInp

delta 47
zcmdn(vfE{XBL|O|ti`@dB5Njkx=9!s>AIyR=^7ar87LT\n",
+    "  
\n",
+    "  \"mass-spring\n",
+    "  
Figure: System with two masses connected by springs
\n",
+    "  
\n",
+    "\n",
+    "First, we need to write the equations of motion, based on doing a free-body diagram on each mass:\n",
+    "\\begin{align}\n",
+    "m_1 \\frac{d^2 x_1}{dt^2} &= -k x_1 + k(x_2 - x_1) \\\\\n",
+    "m_2 \\frac{d^2 x_2}{dt^2} &= -k (x_2 - x_1) - k x_2\n",
+    "\\end{align}\n",
+    "We can condense these equations a bit:\n",
+    "\\begin{align}\n",
+    "x_1^{\\prime\\prime} - \\frac{k}{m_1} \\left( -2 x_1 + x_2 \\right) &= 0 \\\\\n",
+    "x_2^{\\prime\\prime} - \\frac{k}{m_2} \\left( x_1 - 2 x_2 \\right) &= 0\n",
+    "\\end{align}\n",
+    "\n",
+    "To proceed, we can assume that the masses will move in a sinusoidal fashion, with a shared frequency but separate amplitude:\n",
+    "\\begin{align}\n",
+    "x_i &= A_i \\sin (\\omega t) \\\\\n",
+    "x_i^{\\prime\\prime} &= -A_i \\omega^2 \\sin (\\omega t)\n",
+    "\\end{align}\n",
+    "We can plug these into the ODEs:\n",
+    "\\begin{align}\n",
+    "\\sin (\\omega t) \\left[ \\left( \\frac{2k}{m_1} - \\omega^2 \\right) A_1 - \\frac{k}{m_1} A_2 \\right] &= 0 \\\\\n",
+    "\\sin (\\omega t) \\left[ -\\frac{k}{m_2} A_1 + \\left( \\frac{2k}{m_2} - \\omega^2 \\right) A_2 \\right] &= 0\n",
+    "\\end{align}\n",
+    "or\n",
+    "\\begin{align}\n",
+    "\\left( \\frac{2k}{m_1} - \\omega^2 \\right) A_1 - \\frac{k}{m_1} A_2 &= 0 \\\\\n",
+    "-\\frac{k}{m_2} A_1 + \\left( \\frac{2k}{m_2} - \\omega^2 \\right) A_2 &= 0\n",
+    "\\end{align}\n",
+    "Let's put some numbers in, and try to solve for the eigenvalues: $\\omega^2$.\n",
+    "Let $m_1 = m_2 = 40 $ kg and $k = 200$ N/m.\n",
+    "\n",
+    "Now, the equations become\n",
+    "\\begin{align}\n",
+    "\\left( 10 - \\omega^2 \\right) A_1 - 5 A_2 &= 0 \\\\\n",
+    "-5 A_1 + \\left( 10 - \\omega^2 \\right) A_2 &= 0\n",
+    "\\end{align}\n",
+    "or $A \\mathbf{y} = \\mathbf{0}$, which we can represent as\n",
+    "\\begin{equation}\n",
+    "\\begin{bmatrix} 10-\\omega^2 & -5 \\\\ -5 & 10-\\omega^2 \\end{bmatrix}\n",
+    "\\begin{bmatrix} A_1 \\\\ A_2 \\end{bmatrix} = \n",
+    "\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}\n",
+    "\\end{equation}\n",
+    "Here, $\\omega^2$ are the eigenvalues, and we can find them with $\\det(A) = 0$:\n",
+    "\\begin{align}\n",
+    "\\det(B) &= 0 \\\\\n",
+    "\\det (B^* - \\omega^2 I) &= 0\n",
+    "\\end{align}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "omega_1 =  2.24 rad/s\n",
+      "omega_2 =  3.87 rad/s\n"
+     ]
+    }
+   ],
+   "source": [
+    "clear all; clc\n",
+    "\n",
+    "Bstar = [10 -5; -5 10];\n",
+    "omega_squared = eig(Bstar);\n",
+    "omega = sqrt(omega_squared);\n",
+    "\n",
+    "fprintf('omega_1 = %5.2f rad/s\\n', omega(1));\n",
+    "fprintf('omega_2 = %5.2f rad/s\\n', omega(2));"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We find there are two modes of oscillation, each associated with a different natural frequency. Unfortunately, we cannot calculate independent and unique values for the amplitudes, but if we insert the values of $\\omega$ into the above equations, we can find relations connecting the amplitudes:\n",
+    "\\begin{align}\n",
+    "\\omega_1: \\quad A_1 &= A_2 \\\\\n",
+    "\\omega_2: \\quad A_1 &= -A_2\n",
+    "\\end{align}\n",
+    "\n",
+    "So, for the first mode, we have the two masses moving in sync with the same amplitude. In the second mode, they move with opposite (but equal) amplitude. With the two different frequencies, they also have two different periods:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 42,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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Y4vdvGwqikFtrOLEB1a1AhBBIenHsAZY7t6MggC+c2ID+C+0IJL04MWtsZ5zlzi0URKEjtYZjM4HUJ4z4URJI2jFlgWZoKAgQEQSSduRMPr+CWyiIKltilhFrhdEJt97Yov2CSQJJR5zk5kFBgCggkHRkyi620FAQJZyY5dZ0XyiMDhnxoySQdMQglQcFUUIucVTdCvjDrTf0P3SDQNIUL6nzoCDhk9v7VbcC/jDi0A0CSVM8nHpQECXSyUE6pn3AlFPbCSRNMUjlQUGUSCdt/XdT4qL2l834IRJI+mJpmQcFCd+urYOTBzm6yXjFcn00tUl1Ky6OQNIXS8s8KEj4HHvArXF0k/HcmgErGgSBpDMGqTwoSPicmDWWijNqZzRTJpAEgaQ5RvA9KEj4RlObGLUzWn6mYsR4nSCQNLdr6yCDVO0oSPjSSZuOqbnk68SMGK8TBJLmHHuAXwTtKIgSLCcx11SpmhtKqG5FpwgkrTFr4kFBlJDP15TdOMVy3a03TJlAEgSS/ox4rVaYKEj4nJhFJ8lE+emKQd0jQSDpbzS1iSfTdhRECTpJxjGueyQIJP1xZI4HBVHCiVm54US2sKC6IehUtrAwMbJNdSu6QyAZgAMuPSiIEumknU7aDNwZIT9dkT8v1Q3pDoGkOydmccBlOwqiUG4oMXmwygOB5txaY++MYbNHEoFkBn4FeFAQJZyYlRtKZPbNqW4IVpMtzM/uuU7/l01ciEAyQDppH+H3bxsKotBYKs7Anc7kj8a4wTqJQDLArq2MUJ2HgqiVG0oUy3V+BBoqluuTB6uze3aobsgaEUhmYF2ZBwVRqLXijoFT3Zi4sq4dgWQAxx5Q3QS9UBDl0kl7bGc8W5hX3RA8LzM+N7YzbuhgnUQgGcCJWTyKtqMgOsgNJ5zYAJNJmpA/iNyweSvr2hFIZuBXsAcF0YGcTCKTlDN96qiFQDKDY1tufUl1KzRCQXTgxKyJke0scFCrWK5nCwuzu41PI0EgAeiFzCQWOKji1hqZ8UMTI9tM3HV0IQLJDE5sgBu+HQXRhxOzJka2ZfbN8RMJmVtryD2wRi9kaEcgAehVOmlzgkP4soX53HCib9JIEEimcGyLswnaURDdjKXiYzvjiXsOqG5IVGTG50ZTZi/yvhCBBMAfueHE2M54Zpx+UuAy43PppG3Wu446QSAB8I0cQeK1SYHKjM85sQHTtxwti0AC4KfccMKxLTIpIDKNjD4faBUEEgCfkUkB6e80EgQSgCCQSb7r+zQSBJIp3HpjS19sfPMLBdEfmeSjKKSRIJAABIdM8kVE0kgQSAACJTOJ/UlrFp00EgSSKdzaUn+cVeUXCmIQuT+JTFoDud8oImkkhNigugHoiFtv8Fa6dhTELHLTTOKeA7O7d/Ak0Ql5Tt1oKt5/u19XQQ/JDBxb6UFBjHP2HAfOYO1ANNNI0EMyCM+VHhTEOLnhxJaYldk3Rz9pFTKN+uzU1A4RSAYoluvcve0oiLnkIz+ZtBK31kjcc6Cf3ijRFYbsDODWGtG8OldCQYw2lorP7t7B2N2FIp5GgkACED4nZs3u3pEtzPPu85ZiuZ7ZNxflNBIEkhH2lxd3JQdVt0IjFKQPyHef56cr+emK6raoVyzXs4WFiZFtUU4jQSAZgSkTDwrSH2QmFcv1iGeSTKPZ3TsinkaCQDKCW2PPzXkoSN8gkyZL1WxhoXL3jTxjCQJJf3Lil4u1hYL0mShn0mSpOlWqVu6+UXVDdEEg6a5Yrkdtc9zqKEj/iWYm5acrU6Xq7J4dqhuiEQJJd0zge1CQviQzSQgRkUzKFhaK5Tpp5EEg6Y51sR4UpF85MWs0FRcRyKRsYcGtLZFGFyKQdOfWGoxQtaMgfSwKmUQarYJA0tpkqcov33YUpO/1dyaRRqsjkLS2v7yougl6oSBR0MqkPnvVbGZ8TghBGq2CQNJasVwfTW1S3QqNUJCIcGJWn73+PFIvfl0zAklrnCLqQUEipW8ySfaNSKOLIpD0xXyJBwWJoNFU3LEto+eTZKAyUtcJAklfbLjxoCARZPoah/x0hVUMnSOQ9DVZqjI81Y6CRJPMJBPPcchPV9j92hUCSVNyeIoT21ooSJSZeLbQZKlKGnWLQNIUw1MeFCTizMqkYrmen6mQRt0ikDTF8JQHBUErkzQ/Pqr1fiPVDTEPgaQjhqc8KAgkmUnZwoJ8C4mG3FojM35oYmQbl+saEEg6mipVGZ5qR0HQ4sSs3FAis29OdUOWly3Mz+65jt782hBIOiqWF9lw046CoN1YKj62My53m2olMz6XTtqk0ZoRSNph+6cHBcGFcsMJJzag1QIH2ZjccEJ1QwxGIGknP1PhuLZ2FATLyg0l9FngUCzXJw/y+tdeEUh6mSxVhRB0+VsoCFaizwKH1kIGtc3oAwSSXvaXF3NDdPmfR0GwCrnAIVuYV9uMbGF+YmQbj029I5D0wnyJBwXB6sZUn3QnO/Fcpb4gkDSSLSxwWbejIOjExMj2yYNVJQN3bq2RLSxMjGwP/1v3JQJJI8VyneGpdhQEnVA4cCcH69gD6xcCSRfyaByu7BYKgs7J+ZuQV9wxWOc7AkkXLG72oCDonHzlecjvls3PVNh15C8CSQuTpapjW6zSaaEg6FY6aTu2JXstIeASDQKBpIWpUnWUjn8bCoI1mBjZnp8Jabnd/vIi3SPfEUjqFct1t95gJLqFgmBtnJgVTieJ/doBIZDUy09XWEvWjoJgzXLDiRA6SUxwBoRAUozegAcFQS/kTFKgy+2YPQoOgaQYvQEPCoIejabigR7csL+8yARnQAgklegNeFAQ9C6dtN16gKc2yB1ywX1+lBFIKtEb8KAg6J1c2hDQqJ08XJH92gEhkJShN+BBQeCXdNIO6Gi7/eVFxyaNgkIgKZOfrvAClXYUBH7ZtXVwKpjF38VyfQvdo8AQSGqwj8GDgsBHjj0Q0DSSW2twlQaHQFKDU7A8KAh8FNA0kvxAJpCCQyApwD4GDwoCI7i1Rjo5qLoV/YxAUiBbWKA30I6CwHdObCCIdQ1ObMD3z0QLgRS2/HRlLBWnN9BCQRAEx7aOqHiHLHqxQXUDosWtNfbOVCp336i6IbqgIAiO78vhjtQarPkOFD2kUGUL83uHEkyKtlAQBCTQwxoQEAIpPHLjJ5MlLRQEZtkSs8i5QBFI4WHjpwcFgXHc2pLqJvQzAikkbPz0oCAIVLFc9/3qcughBYxFDSHJz9AbOA8FQaDcWsP3uUnHZs13sOghhSFbWEgnbXoDLRQEgQroZFWZcIG+/S/i6CEFzq01JkvV5tfeorohuqAgCFqxXA/o2Piz54gng/hs0EMKXrYwz9hUOwqCoO0vL+4K5oyfXcmgzhGHIJCCJqfuecdPCwVBCIJ7qWvQr6ONOAIpWBxi7UFBELRAX+oa6OtoQSAFKD9dYeq+HQVBCIIbr5NGU/H8dCW4z48yAiko8pQ2JktaKAjCIXtIwX2+HLWjkxQEAikoTN17UBCEQB4eH+i3cGJWOmlPlY4F+l2iiUAKBFP3HhQE4dg7UxlNbQr6u+SGEvSQgkAgBYKpew8KghCE9m4t2UnKFhaC/kZRQyD5j2MIPCgIwrF3ppIbCum5R3aSAjoSIrIIJJ8Vy/Viuc5kSQsFQTiyhYUw360lO0n5GZbb+YlA8hmvVPCgIAiBfO4JeVg4N5Rwa0tMJvmIQPITr1TwoCAIh5LnHidmjabizCT5iEDyjVtrZAsLEyPbVTdEFxQE4VD43DOWiju2xT5ZvxBIvskW5sMcwtYfBUEIiuW62ueeiZHtcsBQVQP6CYHkj2K57tYbrGxuoSAIR366MrvnOoXPPQzc+YhA8ke2sMDUfTsKghBkxuec2IDyScqxVHxsJ5nkAwLJB5wZ6kFBEAI5SqbJc89oKu7WlphM6hGB1CvODPWgIAiB8qkjDydmMZnUOwKpV5wZ6kFBELRzCzi3abVkRmZStrDA8Q1rRiD1hDNDPSgIQpAtzOeGEhqOCTsxKzeUyOybU90QUxFIPckWFlhI1o6CIGhyIYO2Dz0scOjFumazqboNppLXHMNTLRQEQcsWFtza0uyeHaobshq31pgqVYUQPJx1ix7SGnFmqAcFQdAmS1X900ic25lULNdZdNctAmmNODPUg4IgUMVyfapU1T+NpNaiOzKpKwTSWnBmqAcFQaDkIm9T0kgik9aAQFoLpu49KAiCI9OocveNqhvSNTYndYtA6pp8Dxi9gRYKguCc2wBr6miwzKT8dIVM6gSB1B0l7wHTGQVBcFppZPTjDpnUOQKpO0zde1AQBKQ/0kgikzpEIHWBqXsPCoKA9FMaSWRSJwikLjB170FBEIT+SyOplUnySQ4X4qSGTnEMgQcFQRD6NY1a3FojW5hPJ20e5i5EIHXErTUS9xxofu0tqhuiCwqCIJi7wrsrZNJKGLLrCK9U8KAg8F1E0kiwZ3ZlBNLFyUlIbU8XDh8Fge8mS9WIpJEkM2nyYJVMakcgXVx+ukLPuh0Fgb/y05WpUjU6aSQ5MWt29w76Se0IpItgZbMHBYG/soWFYrlu1jl1fpH9JLfeIJMkAukiWNnsQUHgIyPebxQo+ZJZIQSZJAik1eWnK2OpOL2BFgoCH2XG54QQUU4jSb4/SZzbShFlBNKK3Fpj70xFPrxAUBD4Sr6JnLWakhOzcsMJx7ZkSEcWgbSi/Exl71DCiVmqG6ILCgK/ZMbn0kmbNPLIDSfSSTtxzwHVDVFmg+oGaMqtNSZLVTZ+tlAQ+ELuCR1Nxdk2sCw5QZu458Ds7h0RfPijh7Q8Nn56UBD0jjTqRG44MbYzntk359YaqtsSNgJpGcVyvVhe5J5poSDonVtrZPbN5YYTXEgXlRtO5IYSEcwkAmkZvOPHg4KgR/Lwwz4+MtV3Y6n4xMi2qGUSgeRFb8CDgqBHMo1m91xHGnUlnbRnd++IVCYRSF70BjwoCHpRLNdJozWTxwtl9s1F5LV+BNJ56A14UBD0Qh7gTRr1QmZSRF41SyCdh96ABwXBmvX9q/ZCE53XnxNIz6M34EFBsGakkb8ikkkE0vOmSsfoDbSjIFgb0igIUcgkAul5k6UqvYF2FARrQBoFp+8ziUA6K1tY4JdvOwqCNXBrjcz4IdIoODKTsoWFvlwLTiCdNVmqco51OwqCbrHfKBytteD9l0kEkhDnxqYieJThSigIuiVPBiKNwtGvmUQgCSFEfqYymtqkuhUaoSDoljx+lzQKjXzVbGZfX70/iUASk6WqY1vcSC0UBN3KjM+N8irh0I2l4mM74/30/iQCSUyVqqPM3rehIOiKfPcrS2CUkO+q6Jt3nxNIgr2fHhQEnctPV4QQ7FdTSL7TT/4gTBf1QMpPV/jl246CoHPFcn3yYHV2zw7VDYm63FCiWK5PlqqqG9KrqAfS5MEqs/ftKAg6VCzX5ZYj1Q3B2c1JU6Wq6RtmIx1IzN57UBB0jmO8teLErNxwwvTJpEgH0v7yIrP37SgIOpQZnxvbybI6vaSTtukLHCIdSJOlKndUOwqCTsi5CjmXDq2MpuJubcncBQ7RDSQOI/CgIOiEW2tkCwsTI9tVNwTLkJNJkwdNnUyKbiDtLy/uSg6qboVGKAg6IU9k4MFFW/IEB0M7SdENJIanPCgILkoO1rExQHPyB2TiKvCIBhLDUx4UBJ3IFhaYOjLCxMj2/EzFuKNXIxpIDE95UBBcVLawsHcoQTfaCGcH7mYMG7iLaCAVy3Xuq3YUBKuTBwHQPTJIOmm7tSWzVjdENJDcWoPhqXYUBKvLT1c4lMEsTswaTcXNWt0QxUCS8yWqW6ERCoLVFct1jtw1UTppu/WGQZ2kKAYS8yUeFASrmyodo3tkIjmTNFU6prohnYpiIDFf4kFBsDq2BJgrnbTpIemrWK4zX9KOgmB1bAkwmhOz0knblJmkyAWSW2swFN6OgmB1+ZkKbyQx2mhq0+RBMzbJRi6QmC/xoCBYnVtrMF5nNMcecGtmLG2IXCAxX+JBQbAKVmD2ASdmpZOD+x9ZVN2Qi4tcIDFf4kFBsAo60P0hN5ygh6SdYrnOL992FASrowPdHxx7wK0bcK5dtAKJ0XAPCoLV0YHuD07MMuKg1WgF0pFaw7G5u55HQbCKYrmeZryuX6STg/qP2kUrkNx6YwuP7t9lUQAAC7RJREFUe20oCFbh1hpObEB1K+APJzagfycpWoHElIkHBQGgj2gFkhDCsXniOw8FwUqOaP9Ajc45tqX/DzRagcQMrQcFweqYYkSYIhRI/PL1oCC4KCPWCqMTRkwYRymQ6ks87rWjIFjdlpjl1pZUtwIREqVAYsnQ+SgIVkcHup+4tSX9f6ARCiQAXTFlez864dYb+q9gIpAALE9u79d/8wo6YcScMYEEYEXp5KBbZxrJeKac2h6hQOKYHA8KgotKJ20jXluA1e0vm/FDjFAgAejWrq2DprxsFKsolutGvPaXQAKwIoNeNopVmHKuP4EEYEVOzNo7lGDUzmimTCAJAgnA6hi1M11+pmLEeJ2IVCBtiVlsqmhHQdCJdNJ2bItRO0NNlqqObRkxXiciFUgA1iY3nMhPV1S3AmsxVaqOGjJeJyIVSA4Hc52PgqBD8sgGOknGKZbrbr1hygSSiFQgAVgbJ2blhhLZwoLqhqA7+enKxMg21a3oQoQCiYO5PCgIOjeWijOTZJbJUlUIYcrskRShQALQi9wwnSST5GcqueGE6lZ0J0KBJE+KVN0KjVAQdCWdtNNJm0wyQn66In9eqhvSnQgFkhDCiTHmcB4Kgq7khhJubYlrRnPFcn3vjGGzR1LEAomzRM9HQdAVJ2aNpuJ0kjSXn67M7rlOdSvWImKBFBvgEJR2FATdGkvF00mbbUnakj8a4wbrpIgFks3ZBOehIFiD3FCiWK4zcKehYrk+ebA6u2eH6oasUbQCadfWQbaCtqMgWAMnZk2MbM8WFlgUoxW31siMHzJx6qglWoHEzhsPCoK1ObdVdl51Q/C8bGF+71DC0ME6KWKBFLOEEAw1tFAQrJmcTGKBgybkD8K4jUce0QokwbqyC1AQrNloKu7WlljgoNxkqerWlsydOmqJXCClkzbrytpREKyZnExigYNaxXI9P1PpgzQSEQykXVsHuXnaURD0ggUOavXBQoZ2kQskpvE9KAh65MSs2d07MvvmyKSQubVG4p4Ds3uuM3ohQ7voBVLM4tDidhQEvZOL7sikMLm1RrYw309pJCIYSIJZkwtQEPRuLBWXmaS6IVGRLcyPpuL9lEYimoHErIkHBYEvxlLxsZ3xxD0HVDek/2XG50ZTcYNeBduhKAYSsyYeFAR+yQ0nyKSg9WsaiYgGErMm56Mg8JHMJDbMBiQzPufEBvoyjUQ0A0kIMZqKs5uvHQWBj3LDCce2yCTfyTTqm0XeF4poIKWTNoNU7SgI/EUm+a7v00hENpAYpPKgIPCdzKTMOOvufBCFNBKRDSQhRDppT5WOqW6FRigIfJcbTqSTNmscepQZn0sn7b5PIxHlQBpNxekQtKMgCALr7nok19SZfox3h6IbSAxSeVAQBKSVSZzj0BW31ujjFd7Lim4gCZaWXYCCICAykzhbqHPyZKBIpZGIeCCxtMyDgiA4ueEE5911SKZRbjgRqTQSEQ8kOUg1WaqqboguKAgC1TrvjkxahTzDW64HUd2WsEU6kIQQueFEfoZBqudREARqLBXnXRWr6L83SnQl6oHk2ANCCGbyWygIgibfn5QtzHOZeRTL9cy+ucimkSCQnJg1tjPO/psWCoIQyPfM5qcrLKJpKZbr2cLCxMi2yKaRIJAE+28uQEEQAplJxXKdTBKk0TkEEjP5XhQE4SCTpMlSNVtYqNx9Y8TTSBBIEjP5HhQE4SCTJkvVqVK1cveNqhuiBQJJCCHSSZtDCtpREIQmypmUn65Mlaqze3aoboguCKSzcsOJCN4Pq6AgCI3MJCFEpC65bGGhWK6TRu0IpLNY7uxBQRAmJ2aNpuIiMpmULSy4tSXSyINAOkveDxG5GTpBQRCyVib1/Wv9MuNzpNGyCKTnyZPc6BO0UBCETGZSf79qVr5qjzRaFoH0PCdm5YaYOHkeBUH4nJjVx68/j8iLX9eMQDqP3AdAn6CFgkCJvswk0uiiCKTzMHHiQUGgSp9lEmnUCQLJi4kTDwoCVfomk0ijDhFIXkyceFAQKNQHmUQadY5AWgZ9Ag8KAoVkJhn6SCSjlDTqEIG0DPoEHhQEahm6Z5bdr90ikJbH6jIPCgKF5OIas867y09XSKNuEUjLkzeA0SPX/qIgUMusM1iL5Trn1K0BgbSisVScE6/bURCo1cokzS9C+bY90mgNCKTV5IYT9AnaURCoJTMpW1hwaw3VbVmeW2tkxg+ximFtCKTVyNcC8e7UFgoC5ZyYNTGyLbNvTs9Myuybm91zHe9+XRsC6SImRrbz7tR2FATKpZP22M54tjCvuiFemfG5sZ1x0mjNCKSLcGJWOmkzTtVCQaCD3HAinbS1WuAgG5MbTqhuiMEIpIvLDSX0n0cNEwWBDuRCcE0GkFlW5wsC6eLYFupBQaADucAhP1NRPpkkFzLQN+odgdQRtoV6UBDoQD4bKZ9MyhbmJ0a2MXXUOwKpI/KlYUyctFAQaGJM9alC8lvLZqBHBFKn5IpnxqlaKAg0MTGyffJgVUl/3a019s5UJka2h/+t+xKB1AV53SsfsNYHBYEOFE5qysE6J2aF/637EoHUBSdmje2MswunhYJAE3L+JuQVd/LbMVjnIwKpO3KlKZP5LRQEOmituAvzm+ZnKqys8xeB1B15bAmT+S0UBJpwYlaY51pNlqrppM3KOn8RSF1jMt+DgkATYXaS8jOV0dSmcL5XdBBIa8FkvgcFgQ5C6yTRPQoIgbQWTOZ7UBBoIjecCOE6pHsUEAJpjUZTcbe2xGR+CwWBDuQAcqDX4WSp6tgW3aMgEEhrxFEFHhQEmhhNxQOd0dxfXhxlqXcwCKS1k4PITOa3UBDoIJ203XqA05lyAim4z48yAqknuaEEk/ntKAiUk0sbAhq1myxVx1JxjmYICIHUE00OG9YHBYEO0kl7/yOLQXzy/vKiY5NGQSGQeqXkzBKdURAot2vr4OTBQK7AYrm+a+tgEJ8MQSD1TsmZJTqjIFDOsQcCGjd2aw0mkIJDIPnAiVnppM0CsxYKArWcmOXE/J9GYnI0aASSP3JDCXbhtKMgUCuImZ5iuc7Z3oEikPzhxKzRVJw+QQsFgVpOLKhROwSHQPLNWCoe5mHD+qMgALpCIPlJTubzXNZCQaCKY1tH/L7wjtQarPkOFIHkJ84Y9aAgADpHIPmMM0Y9KAiUcOuNLX6fp7AlZgV6KBEIJJ9xxqgHBUHfcGKWW1tS3Yp+RiD5jzNGPSgIwlcs14PYwUoPKVAEUiA4Y9SDgiBkQVxsjj3g+2eiHYEUCM4Y9aAgCJOcs/T9TO5AzxGHIJCCI4cLuHZbKAhC49YawR2pENA54hAEUnCYzPegIAjN/vLirmQgZ3KPpuI8VAWHQAqQnMznV3ALBUE4gnupa9Cvo404AilYuaFEsVznkaqFgiBogb7UlWmkQBFIwZKT+ax4bqEgCFpw43XSaCrOBRwQAilwTOZ7UBAESvaQgvt8OWrHBRwEAilwTOZ7UBAEJ1tYCPqVRfL9k1OlY4F+l2gikMLAUQUeFAQBmSxVc0OJoL+LnAoN+rtEEIEUEo4q8KAg8F1+uhLccoZ2spNEL993BFJIeBGDBwWBv9xaY+9MJYTukSQ7STxR+YtACo/cUkdPv4WCwEfZwvzeoUQI3SPp7HpRnqh8RSCFR17BzIW2UBD4xa013HojNxxS90hKJ22eqPy1rtlsqm4DAAD0kAAAeiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFogkAAAWiCQAABaIJAAAFr4/7aJQLdZl37BAAAAAElFTkSuQmCC\n",
+      "text/plain": [
+       ""
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "t = linspace(0, 3);\n",
+    "subplot(1,5,1)\n",
+    "plot(sin(omega(1)*t), t); hold on\n",
+    "plot(0,0, 's');\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "box off; set(gca,'Visible','off')\n",
+    "\n",
+    "subplot(1,5,2)\n",
+    "plot(sin(omega(1)*t), t); hold on\n",
+    "plot(0,0, 's');\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "text(-2.5,-0.2, 'First mode')\n",
+    "box off; set(gca,'Visible','off')\n",
+    "\n",
+    "subplot(1,5,4)\n",
+    "plot(-sin(omega(2)*t), t); hold on\n",
+    "plot(0,0, 's');\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "box off; set(gca,'Visible','off')\n",
+    "\n",
+    "subplot(1,5,5)\n",
+    "plot(sin(omega(2)*t), t); hold on\n",
+    "plot(0,0, 's');\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "box off; set(gca,'Visible','off')\n",
+    "text(-2.7,-0.2, 'Second mode')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can confirm that the system would actually behave in this way by setting up the system of ODEs and integrating based on initial conditions matching the amplitudes of the two modes.\n",
+    "\n",
+    "For example, let's use $x_1 (t=0) = x_2(t=0) = 1$ for the first mode, and $x_1(t=0) = 1$ and $x_2(t=0) = -1$ for the second mode. We'll use zero initial velocity for both cases. \n",
+    "\n",
+    "Then, we can solve by converting the system of two 2nd-order ODEs into a system of four 1st-order ODEs:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Created file '/Users/niemeyek/projects/ME373-book/content/bvps/masses.m'.\n"
+     ]
+    }
+   ],
+   "source": [
+    "%%file masses.m\n",
+    "function dxdt = masses(t, x)\n",
+    "% this is a function file to calculate the derivatives associated with the system\n",
+    "\n",
+    "m1 = 40;\n",
+    "m2 = 40;\n",
+    "k = 200;\n",
+    "\n",
+    "dxdt = zeros(4,1);\n",
+    "\n",
+    "dxdt(1) = x(2);\n",
+    "dxdt(2) = (k/m1)*(-2*x(1) + x(3));\n",
+    "dxdt(3) = x(4);\n",
+    "dxdt(4) = (k/m2)*(x(1) - 2*x(3));"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 65,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       ""
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "clear all; clc\n",
+    "\n",
+    "% this is the integration for the system in the first mode\n",
+    "[t, X] = ode45('masses', [0 3], [1.0 0.0 1.0 0.0]);\n",
+    "subplot(1,5,1)\n",
+    "plot(X(:,1), t); \n",
+    "ylabel('displacement (m)'); xlabel('time (s)')\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "%box off; set(gca,'Visible','off')\n",
+    "\n",
+    "subplot(1,5,2)\n",
+    "plot(X(:,3), t); xlabel('time (s)')\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "text(-4,-0.2, 'First mode')\n",
+    "\n",
+    "% this is the integration for the system in the second mode\n",
+    "[t, X] = ode45('masses', [0 3], [1.0 0.0 -1.0 0.0]);\n",
+    "subplot(1,5,4)\n",
+    "plot(X(:,1), t);\n",
+    "ylabel('displacement (m)'); xlabel('time (s)')\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "%box off; set(gca,'Visible','off')\n",
+    "\n",
+    "subplot(1,5,5)\n",
+    "plot(X(:,3), t); xlabel('time (s)')\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "text(-4,-0.2, 'Second mode')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This shows that we get either of the pure modes of motion with the appropriate initial conditions.\n",
+    "\n",
+    "What about if the initial conditions *don't* match either set of amplitude patterns?"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 56,
    "metadata": {},
-   "outputs": [],
-   "source": []
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       ""
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "[t, X] = ode45('masses', [0 3], [0.25 0.0 0.75 0.0]);\n",
+    "subplot(1,5,1)\n",
+    "plot(X(:,1), t);\n",
+    "%plot(0,0, 's');\n",
+    "set (gca, 'ydir', 'reverse' )\n",
+    "%box off; set(gca,'Visible','off')\n",
+    "\n",
+    "subplot(1,5,2)\n",
+    "plot(X(:,3), t);\n",
+    "set (gca, 'ydir', 'reverse' )\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In this case, the resulting motion will be a complicated superposition of the two modes."
+   ]
   }
  ],
  "metadata": {
diff --git a/content/bvps/finite-difference.ipynb b/content/bvps/finite-difference.ipynb
index f0495c2..e8fc766 100644
--- a/content/bvps/finite-difference.ipynb
+++ b/content/bvps/finite-difference.ipynb
@@ -631,20 +631,8 @@
     "\\frac{d^2 T}{dx^2} - \\frac{h P}{k A_c} \\left(T - T_{\\infty}\\right) - \\frac{\\sigma \\epsilon P}{h A_c} \\left(T^4 - T_{\\infty}^4 \\right) = 0\n",
     "\\end{equation}\n",
     "\n",
-    "This is a bit trickier to solve because of the nonlinear term involving $T^4$. But, we can handle it via the iterative solution method discussed above, moving the nonlinear parts to the right-hand side:\n",
-    "\\begin{align}\n",
-    "\\frac{T_{i-1} - 2T_i + T_{i+1}}{\\Delta x^2} - m^2 \\left( T_i - T_{\\infty} \\right) - M^2 \\left( T_i^4 - T_{\\infty}^4 \\right) &= 0 \\\\\n",
-    "\\frac{T_{i-1} - 2T_i + T_{i+1}}{\\Delta x^2} - m^2 T_i &= M^2 \\left( T_i^4 - T_{\\infty}^4 \\right) - m^2 T_{\\infty} \\\\\n",
-    "T_{i-1} + T_i (-2 - \\Delta x^2 m^2\n",
-    "\\end{align}"
+    "This is a bit trickier to solve because of the nonlinear term involving $T^4$. But, we can handle it via the iterative solution method discussed above."
    ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {
diff --git a/content/bvps/masses.m b/content/bvps/masses.m
index 006138d..6d4d929 100644
--- a/content/bvps/masses.m
+++ b/content/bvps/masses.m
@@ -1,4 +1,5 @@
 function dxdt = masses(t, x)
+% this is a function file to calculate the derivatives associated with the system
 
 m1 = 40;
 m2 = 40;
diff --git a/content/bvps/shooting-method.ipynb b/content/bvps/shooting-method.ipynb
index b6c9e01..1ba9df2 100644
--- a/content/bvps/shooting-method.ipynb
+++ b/content/bvps/shooting-method.ipynb
@@ -75,7 +75,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [
     {
@@ -91,7 +91,7 @@
     },
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        ""
       ]
@@ -178,7 +178,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [
     {
@@ -244,7 +244,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
@@ -275,7 +275,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
@@ -336,7 +336,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
@@ -406,12 +406,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
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ioqJefvllEamsrLRYLKtXr/7qq6+Ki4u1cJo0aZKItGrVKjIycsuWLR4sHQCMQUsjs9Xmy2kk9Zj2HRMTY7VaazwUGBgYGRm5fPny5OTkzMzMxMTEcePGaU8lr6ioOHz48IMPPtjQegHAWMyltrg3081Wm2+OG7m74hbSzJkzH3744S+++KL20wICAiIiIhYvXrx48WK73V5YWLhhw4Z77rmnvnUCgAGZS209ln0dGRqcP/9mvWvR3xW3kEJCQuLi4t588826vyUgIKB79+7z58/v16/flX4cABjVnlxrj2VfD48OIY009Vmp4ZlnnjGbzQsWLPB4NQDgI1yrpvrOykCXVZ9AevbZZz/99NOPPvqof//+M2bMMJvNlZWVHq8MAIwq/kARaVRdfe5D2rx5s7Zts9m++OILbTypY8eOsbGx06dP7969e0BAgIfLBACjWPKv/MWJ+ROHdDH241/roT73IW3ZsqV3795lZWU5OTlLly61WCwVFRUlJSXbt2/fvn27iHTu3Pmuu+565plnGqFgAPBikzYdiT9QtHhkj0Wjeuhdi3KuuMtOW6PB398/NDQ0JiZm+/bt6enpqampCQkJvXr1atWqlYgUFxe/8847BQUFjVAwAHiruFXp8QeK1j/cmzSq0RW3kN5+++0hQ4b8/ve/nzJlSnh4uLYzJCRECycRKSsry87OXrduXUREhIeLBQCv5ePLAtVFfSY1HDhwYNCgQWPHjh08eHBWVlaVo1o4vf32254oDwC8nnazEWl0WfV8/MTIkSNHjhzpcDj8/XnEHwBckjaFQbv11aceblQPDXpAH2kEAJfiWi+VKQx1xBNjAcDzXA0juunqjkACAE+iYVRvBBIAeEz8gaJJm47QMKofAgkAPMDVMGIJhnojkACgobSVUmkYNRCBBAD1Zy61vXugaHFiPiulNhyBBAD15GoYMX/BIwgkAKgPbWI3DSMPIpAA4MqYS21xb6abS200jDyLQAKAK8BSQI1H9UByOp1+fn56VwEA3PHa6BQNpE8//TQ5OfnIkSMFBQXh4eFRUVEjRowYN25cs2aKFgzAwFxT6ZjY3aiU+/1utVrnz5+/a9cu156SkpKSkpL9+/cnJCS8+eab0dHROpYHwKe4RxENo8amXCDNnj07JSVFRFq3bj1s2LBevXodP378s88+O3/+fEFBwYwZMz766KMWLVroXSYA43MNFxFFTUOtQPrmm2+0NAoPD1+/fn2vXr20/VOnTv3DH/5QXFx87NixDz74YOLEiXpWCcDotCgSEaKoKakVSB988IG2MWfOHFcaiUj37t0XLVo0depUETl06JA+xQHwAXtyrZM2HTGX2oZHh6x/uA/z6JqSWoFUVFQkIq1bt77rrruqHLrpppu0jSNHjjR1WQB8gGsSHVO69aJWIBUXF4tI165dAwMDqxyqqKjQNlq1atXUZQEwNHOpbUlifvyBIibR6UutQHrjjTd++eWXtm3bVj+0f/9+bcO9Kw8AGoJJdEpRK5D69etX4/7y8vKXXnpJ2x42bFgdr2axWNLS0lwvTSaTyWRqYIUAjMFHosj9d2BqaqqOldSFWoFUo8LCwj/+8Y8lJSUiEhMTM2rUqDq+ce7cue4vExISCCQA4jOT6NLS0iZMmKB3FVdA9UD64osvlixZUlpaKiJRUVGvvPJK3d87ffr02NhY18uYmBjP1wfAq8QfKFqSmO8j66KaTKaEhATXS4vFUuXPdNWoG0inTp1avHjxl19+qb2MiYlZuXJl+/ZXMNgYGxtLCAHQuM/n3j11oC9MoqsyTuHefacmRQNp27ZtL7zwwi+//CIiwcHBM2bMmDhxor+/v951AfA+7vO5mUSnMhUD6eWXX167dq22PWLEiPnz53ft2lXfkgB4I/eZC+sf7j1xSBe9K0JtlAskVxq1bdt20aJFY8aM0bsiAN7H/dYiXxguMga1AikzM/Odd94RkfDw8A0bNrCwN4ArteRf+fHfFplLbUSR11ErkFasWOFwOETk//7v/0gjAHXn6p0TEd+ZtmAwCgXS2bNnDxw4ICItWrQ4evTo0aNHazytW7duQ4cObdrSAKiryi2ujw7pQhR5KYUC6ccff9SaR+fOnVu0aNGlTvvNb35DIAEQBooMR6FAOnbsWF1Oa968eWNXAkBxDBQZkkKBNH78+PHjx+tdBQB1VRkoWv9wb24qMhKFAgkALsU9ihgoMioCCYDSGCjyHQQSAEUxUORrCCQAanHvnWPJH59CIAFQBQNFPo5AAqA/BoogBBIAHWlNIgaKoCGQAOiAgSJURyABaDo0iVALAglAU3CNEonI4pE9hvUMYZEFVEEgAWhEVbrmaBKhFgQSgEax5F/5e3Kte3LLeCoE6ohAAuBJNIlQbwQSAA+oPluBJhGuFIEEoEHcm0QTh3QZNjKECdyoHwIJQH0wgRseRyABuDLxB4qSc8u0ZX4mDu5C1xw8hUACUCfMVkBjI5AA1IbZCmgyBBKAGmg5pN1IJMxWQJMgkAD8h5ZDZqtNW+OHrjk0JQIJgAhDRFAAgQT4tOo5xBAR9EIgAb6IHIKCCCTAh1SfMsdjIKAOAgkwvio5NHFwF3IICiKQAMOqMnV7eHTIxMFdmKoAZRFIgAG5VvcRpszBexBIgHEwdRtejUACvB5T5mAMBBLgrcghGAyBBHiZPbnW5Jwy9xxiyhyMgUACvEP1HKI9BIMhkAB1aZ1yIkIOwRcQSIByalxymxyC4RFIgCqq3MfKvG34Gi8IJLvdLiIBAQF6FwI0CnII0KgeSCkpKZMnTxaRgwcPtmzZUu9yAI8hh4AqVA+k999/X+8SAE+qvr4cOQRo1A0kh8Px4osv7tq1S+9CAA+ost52ZHvaQ0BVygXSDz/8cOjQodzc3MTExJMnT+pdDtAg1Z/7EBEaPHFIF73rAlSkXCC9/PLLqampelcBNAg5BNSDcoEUGhrauXNn18tffvnl3LlzOtYD1JG51Ga2nnNfTIHn4AFXRLlAeu2119xfzpkzZ+vWrXoVA1xWjZPlyCGgHpQLJMAr1JhDLKYANISRA8lisaSlpblemkwmk8mkYz0wACbLwbu4/w5Uf3jeyIE0d+5c95cJCQkEEuqBwSF4qbS0tAkTJuhdxRUwciBNnz49NjbW9TImJkbHYuB1GByCtzOZTAkJCa6XFoulyp/pqjFyIMXGxhJCuFIMDsEwqoxTuHffqcnIgQTUHYNDgO4IJPguBocApRBI8DkMDgFqIpDgKxgcAhRHIMHgGBwCvAWBBAPSQshstcUfKBIGhwAvQSDBOOiUA7ya6oHUvHlzvUuA6vbkWt1nytEpB3gp1QPphRdeeOGFF/SuAsqp3inHTDnA26keSIA7OuUAAyOQ4AWqdMoNj26/OLo9nXKAwRBIUBS3rwK+hkCCWtxvGxI65QBfQiBBf1oIiYj7mnIiQqcc4FMIJOiGTjkA7ggkNDVuGwJQIwIJTYHGEIDLIpDQiJihAKDuCCR4WI0zFCJCgycO6aJ3aQCURiDBM+iUA9BABBLqr/ojwJmhAKDeCCRcMRpDABoDgYS6Yro2gEZFIKE2NIYANBkCCTVgujaApkcg4d+qP/Ju4uAuNIYANBkCydfRGAKgCALJF7G6NgAFEUg+hBkKAFRGIBlcjY2h4Tz/G4B6CCRjojEEwOsQSMZhLrXtybUWlNpcjaHh0e0X0xgC4CUIJK9HYwiAMRBIXql6Y4iFfAB4OwLJm9AYAmBgBJLqeMQDAB9BICmKxhAAX0MgKYTGEABfRiDpj8YQAAiBpJcqjSERGR4dwqqmAHwZgdSkaAwBwKUQSI2OxhAA1AWB1FhoDAHAFSGQPInGEADUG4HkATSGAKDhCKT625NrTc4pcz38m8YQADQEgXRlqjeGJg7uQmMIABpO0UDau3fvxx9/nJOTc/LkycDAwJ49ew4aNGjKlCktW7bUpZ4qjSGtU47GEAB4kHKBZLfbFyxYsGXLFvedp0+fTk1N3bp168qVK6+77rqmqYTGEAA0JeUCKT4+3pVGt956a//+/UtKSpKSkk6dOmWxWGbNmrV9+/bg4EZsl9AYAgBdqBVIZWVlK1euFBE/P79ly5bdf//92v7p06c//vjjx44dO378+Pr166dOnerZz6UxhMaTlpZmMplMJpPehQCqUyuQkpOTbTabiMTFxbnSSEQ6deq0cOHCRx55REQSExM9FUg0htDY0tLSJkyYkJCQQCABl6VWIKWkpGgbY8eOrXIoNjY2IiKioKAgMzPz9OnTYWFh9fsIGkMAoCa1AqmgoEDbuOGGG6ofveGGG7QTjh8/fqWBRGMIABSnViAVFhaKSJs2bdq3r6G9EhERoW1YLJYBAwZc9mrm0nP//Fe+qzEUePb0iK6OqYNChvfULn6yJOdkieeKB6pLTU0VEYvFkpaWpnct8HXat1FlagVSRUWFiNSYRu77tdNqdzas10Nf2kXyA8+eDvvpm7Yn9geeO50rkiuyzoMVA3Uwd+5cvUsAvIBagXThwgURad68eY1HAwMD3U+rhclk2vzawrNhV7c8fUyktUiEyMOeLRUAvFFMTIzeJVySWoEUGBhot9u1iXbVORwObcPpdNZ+nf/Mso1W958eAODOX+8C/kvtSeNqGLmaSgAAw1ArkIKCguTSQ0Rnz551Pw0AYCRqBVKnTp1ExGq1lpWVVT+al5fnfhoAwEjUCiTX3ezZ2dnVjx47dkzb6NatW9PVBABoEmoF0uDBg7WNbdu2VTmUnZ199OhREQkPD7/qqquaujIAQCNTK5Buv/12bWPz5s0//vij+6GlS5fa7XYRiYuL8/Pz06E4AEBjUiuQoqOjR48eLSIOh2PatGnbtm0rKCjIyMiYOXOmdqN7cHDwlClT9C4TAOB5fpe9p6eJnT59+oEHHtDWEKrCz89v4cKF48ePb/qqAACNTa0WkoiEhYVt37593LhxLVq0cO308/Pr1avXe++9RxoBgFEp10ICAPgm5VpIAADfRCABAJRAIAEAlEAgAQCUQCABAJRAIAGAr7Db7dqSN2pS6wF9Dbd3796PP/44Jyfn5MmTgYGBPXv2HDRo0JQpU1q2bKl3aTCUhn/TFixYUOOq9prZs2dHRER4qFhARCQlJWXy5MkicvDgQTV/JRrnPiS73b5gwYItW7ZUP2QymVauXHndddc1fVUwHo98086ePTtw4MBa/vd9+OGH/fv3b1ChwH+bNm3arl27ROFAMk4LKT4+3vU74tZbb+3fv39JSUlSUtKpU6csFsusWbO2b98eHBysb5EwAI9807Kzs7U06tevX7t27aqfEBYW5vHK4bMcDseLL76opZHKDNJCKisrGzZsmM1m8/PzW7Zs2f3336/t//nnnx9//HHtQUozZ86cOnWqrmXC63nqm/bBBx8899xzIpKUlOR6DBjgWT/88MOhQ4dyc3MTExNPnjzp2q9sC8kgkxqSk5NtNpuIxMXFuX5HiEinTp0WLlyobScmJupTHAzEU9+0rKwsEWnTpg1phMbz8ssvP//88xs3bnRPI5UZJJBSUlK0jbFjx1Y5FBsbqw0OZ2Zmnj59uqkrg7F46pumPRP52muvbYQagX8LDQ3t7MZ9xWo1GWQMqaCgQNu44YYbqh+94YYbtBOOHz9O1zwawlPfNK2F1Lt3bxEpKSnJy8tr06ZNVFSU+r8y4EVee+0195dz5szZunWrXsXUhUECSXt+Ups2bdq3b1/9qGv6rMViGTBgQJNWBmPxyDetqKiovLxcRKxW6+jRo/Pz87X9/v7+/fr1W7hwYb9+/TxfOqA8g3TZVVRUiEiNvyPc92unAfXmkW+a1jwSke3bt7vSSEQcDsf3338/duzYd955xzPlAl7FIC2kCxcuiEjz5s1rPBoYGOh+GlBvHvmmuQLJ399/7Nixt9xyS+fOnY8dO7Z27dq8vDyHw/Hqq68OHTq0V69eHq0dUJ1BAikwMNBut2vTn6pzOBzahjHmuENHHvmmlZaWhoSEOJ3Ol156KS4uTtt5/fXXjxkzZubMmUlJSRcvXly2bNm7777r2eIBxRmky672//+uP1ddf8AC9eORb9q8efNSU1PT0tJcaaQJCgpaunRp69atRSQjI8MVb4CPMEggBQUFyaU77s+ePet+GlBvjf1NCw8P12ZD2Gy2HR6g0wAAEFVJREFUn376qX4XAbyUQQKpU6dOImK1WmtcrTIvL8/9NKDemuCb1rVrV23j+PHj9b4I4I0MEkiu2921+w2r0BZ0EZFu3bo1XU0wooZ/08rLy5999tm5c+du27atxhMsFou2wd9P8DUGCaTBgwdrG9X/k2dnZx89elREwsPDr7rqqqauDMbS8G9aUFDQ1q1bP/nkk/Xr11c/Wl5efujQIREJDAzs0aOHx+oGvIFBAun222/XNjZv3vzjjz+6H1q6dKn2QKq4uDg/Pz8dioOBXOk3bceOHZs2bdq0adPBgwe1PUFBQdoCDZmZmRs3bnS/gtPpXLFihXbP7PDhw5s1M8gkWKCOAhYvXqx3DR4QGhqak5OTk5PjdDqTk5PDwsKCg4PNZvOKFSuSk5NFJDg4+JVXXgkJCdG7Uni3K/2mTZ8+/dNPP92zZ0/btm2HDh2q7WzTps0XX3whInv37j1+/Hjbtm3tdvv333+/cOHCnTt3ikiLFi3eeuutNm3a6PRTwpi+/PJL7R64J554Qs0px8b5E+y55547dOhQYWHhzz//PHv2bPdDfn5+c+bM4fmb8IiGf9PuvPPOjIyM+Ph4p9O5devWKsuLtW3bdvny5a6pDYDvMEiXnYiEhYVt37593Lhx7stT+vn59erV67333hs/fryOtcFIruib5lrTocofpPPmzVu7dq3Wd+fSrl27UaNG7dix44477mi08uG7LrXCiDoM8oA+d06n02KxmM3mkJCQ6Oholk9GI/HIN+3XX3/Nz8//9ddfo6OjO3fu7PEiAS9iwEACAHgj43TZAQC8GoEEAFACgQQAUAKBBABQAoEEAFACgQQAUAKBBABQAoEEAFACgQQAUAKBBABQAoEEAFACgQSlJScnux5tB8DYjPM8JBhPcnLyH//4RxH59NNPqzypAYDx0EKCurTHqopI69at9a0EQBMgkKCur7/+WkSGDh3avXt3vWsB0OgIJCjKbDYXFxeLyOOPP653LQCaAoEERaWmpopIRETEzTffrHctAJoCkxqgv5ycnLy8vNLS0tDQ0KFDh7Zs2VJEvvrqKxEZN25cjW8pKipaunSpiEyaNGnIkCF1/KCMjIzVq1c3a9Zs+fLlBhiXWrZsmcViMZlM8+fPz8vLs9vtIhIVFRUQEFDLu86ePWuxWETEz8+vZ8+eTVQrUAcEEvRktVrnzJmzd+9e156EhISYmBiHw5GcnNysWbMHH3ywxjcuWbJk9+7dHTp0uO666+r+cT169EhLS/v111+7du06d+7chlavq7y8vA0bNojIhAkTRGT8+PGlpaUisnLlylGjRtXyxvT0dK0XtFmzZocPH26SYoE6ocsOelq0aJF7Grmkp6fbbLb777+/xnbM559/vnv3bhH585//3KJFi7p/XLt27R577DERSUhIyM/Pr2/VSvjoo4+0jYceesh9v9ZOArwRgQTdnD9/PikpSdu+55573njjjVdffbVPnz4isn//frlEf11FRcWyZctEpEePHg888MCVfujEiRNDQ0MvXry4fPnyBlWvq8rKyk8++UREbrjhhquvvlrvcgDPIJCgm+Li4srKShFp1arV888/f8cdd4wZM0ZrEu3fv3/AgAE13gz76aefnjp1SkQmT55c+2BJjVq2bDl+/HgR2bt375EjRxr6M+hk586dVqtVRMaOHat3LYDHEEjQjcPh0DYiIyPde94qKyvT09Or9ERpnE7ne++9JyJBQUGjR4+u3+f+9re/1Tbefffd+l1Bdx9++KGItG7d+q677tK7FsBjCCTooLCw0Gw2a3O9ROTChQtms9lsNv/0008ikp2dHRUVde+991Z/49dff52XlycicXFx9Z4mFxERcf3114vIjh07tMZWEzt//rz285aXl9d4QnFxsdlsLigoqPHoiRMnvvnmGxH57W9/e0VDaIDimGUHHUybNs29u+zYsWPaxLCIiIjExMS+fft+9tlnNb5x+/bt2sbdd9/dkALuvvvuQ4cOXbhw4bPPPps4cWJDLlUPR44c0dp/w4YNW716dfUTnnnmGW0UrcZF/D7++GOn0yn018FwaCHBm2h3y4rIwIEDG3Id19u1u530cvHixSs9wW63b968WUT69u2rTQABDIMWEnQwceJEi8VSXFysjYV06NBBm1DXrl27Wt5VWFhYWFionR8eHu5+6NSpU2+88caZM2duv/1217BKXl7e1q1bT5w44e/vf9NNN91///2u86+55pqAgAC73X7gwIHKysrAwECP/4yNZPfu3SdPnpRqs70BAyCQoIP77rtPRPLz87VA6tix45NPPnnZd6WlpWkbffv2dd/vcDj+8pe/PPXUU+fOnZs8eXJWVtasWbNef/31zMzMhx56aMqUKfv37//LX/6ya9euVatWaW8JCgqKjo7Ozs4+d+5cRkZG7Ws9mM3mX3/9te4/Xd++ff39G6vvQfsXa9my5ZgxYxrpIwC9EEjwGidOnNA2rrrqKvf9H3300ZAhQwYPHqxN23vnnXesVmuPHj3efvtt7YQRI0YMGTJk165d6enprs66qKio7OxsETl+/HjtgbR8+fLk5OS615mRkdFIcw2KiopSUlJE5O67727VqlVjfASgI8aQ4DW0pXFEpG3btu77d+7cqc3k1hLrwoULQUFBkyZNcj9H6wzUevyqXOT06dO1f66fn19DS/eQzZs3a6HLdAYYEi0keA1XILVp08a102635+fna4uEfvvttyJiMpn++te/VnmvtlBQs2b/+cLXPZAWLlw4derUutcZFBRU95PrzuFwaNMZrr32Wm3aOmAwBBK8huuuHffeKofD8be//U3b1qbM3XHHHVUi4dy5c1rv3LXXXuva6bqNyZVzl9KtW7du3bo1tPoG27dvn9bCq715pM0Ir8sJzZs391RtgEfQZQev4YqQc+fOuXYGBga6hoW0SeGxsbFV3qgFVVhYWGRkpGuntmqRKPl7ucZQ0aYzBAcH33PPPdWPulqNP//8c+0XLykpaXCBQKOghQSvERYWpm3UOOetoKDg1KlTAQEBt9xyS5VDBw4cEJEqD/pzXcR12UuprKys+xLafn5+De+yqx4qp06d0haivfPOO917LF2ioqK0lR0uu4q5awEI93gGVEAgwWu4kqPGFXe0AaS+fftWz4P09HQRuemmm9x3ui4SGhpa++c++eSTdZ9lFxAQcPDgwdozyTUp3NVKc3fhwgVtCSV3W7Zs0ULxUv11UVFR2iM5vv32W7vdXsuys99995224d6BCaiALjt4jY4dO2obxcXF1Y9q/XKDBw+usv/8+fPaY+iqHNJuLxWRDh06eLBIu93uWjT2Ulx39R4/frz60ffee6/KAg1Op1N7+lHPnj0vtURFdHS0tpGXlxcfH3+pj/7yyy+15BaRGhdTB3RECwlew5UomZmZ1Y9qA0gxMTFV9n/11Vd2uz08PDwiIsJ9v+siAwYMqP1zn3zyyTvuuKPudV52UKpTp07+/v4Oh6O4uHjbtm3uY0LffPON6+5dl9TUVC26apnOcNddd7311lvaaStXrgwMDBw/fnyVdlJSUtLixYu17dDQUG6thWoIJHiN6OjoTp06/fzzz8ePHy8vL3e/G8k1gDR06NAq79IGkKr01xUVFWnPE+ratWvXrl1r/9z+/fv379/fMz+DiIgEBASMGDEiMTFRRObOnfvdd98NHDjQarUeOnTo888/rz6jQZvO0Lx5c22Fixq1aNFixYoVf/jDHxwOh81mW7Zs2YcffnjLLbdERka2atXKbDZ///33+/btc53/3HPPXbavEmhiBBK8yS233LJlyxYROXz4sHvGaN1Q/fr1q74qnTaAVGWmw48//qhtVJ+S1zS0canz58/b7fZNmzZt2rTJdej2228PDQ39+OOPtZdWq/XLL78UkVGjRtW+1t+gQYOeffbZV199VZuFeOzYsWPHjlU/LTAwcMqUKXfeeacnfx7AExhDgm5cI/91n5Y2cuRIbUNrXrhoo0S33nprlfMrKyt/+OEHqbY6+K5du7SNuLi4KyvaQ6699trNmzdXWZSvY8eODz744Ouvv+7+qKetW7deuHBB6rY6w4QJE3bu3DlhwoRLLRcbFxe3Y8eOGTNmNKx8oFH4XfY2OkAdTqdz1KhRBQUF7du337dvn2vlhbKysu+//37YsGFVzk9KSpo6dWrHjh21JeA0Npvt5ptvrqio6Ny5c1JSUj2eg+5BJ0+ePHr0aFBQkMlkMplM1U/Yt2/foUOHmjdvPnny5LpftrKy0mKxnDhx4sSJEwEBASEhIRERERERETzQDyqjyw7exM/P79FHH33++eetVmtKSoqrfRMSElI9jeT/99fdeOON7juTkpIqKipEZNy4cfqmkYh06NCh9ml+Q4cOrT4wdlmBgYGRkZHcaQTvQpcdvMzvf//7kJAQEdm4ceNlT9YCqcov9Pfff19EgoODWaIUUAqBBC/TokWLmTNnikhKSorrCUk1stvtGRkZIuL+dIm9e/dq75o+fTrTzAClMIYE72O323/3u99lZWUNGDDAfX5aFbt3737iiSc6d+7sWmfB4XDcd999WVlZvXv33rx5s+79dWpKSUnR5vXFxMT85je/qT7s5HA4EhMT9+/f73A4brvtthEjRuhRJgyIFhK8T0BAwIIFC0Tk4MGDBw8evNRpWvPIfQm7PXv2ZGVlBQQELFu2jDSq7uzZs48++ugrr7xy5syZvLy8p59++rbbbtOeeeGSlpY2cuTI/fv333bbbSNHjiwuLp42bdqZM2f0qhlGQgsJ3kpb262WXPn2228/+OCDp556qkuXLtoeh8OhfeFJoxo99dRTQ4YMGTdunPYyPT19+vTpp0+ffuyxx5555hkR2b1793PPPbd69Wr3ZYdycnLeeeed5cuX61M0DCTAtZQI4F38/f1dq5TWqGvXriNHjnRfG9vPz++y7/JZWVlZ+fn5Tz75pGtPly5dbrzxxs8++ywtLS04OLht27Z//vOf33333Wuuucb9jaGhoVlZWWFhYe3bt2/yqmEotJAAiIisWrVq9OjRUVFRVfbv3LlTSymTyTRjxox77723+ntPnDjxxRdfTJkypSkKhXHxpyIAEZGCgoLqaSQiI0aM0O7JLS4urv6sKU337t3z8vIatz74AAIJgIiItgB5jYf+9Kc/iYjdbp85c2aN5zgcjkstVgTUHWNIAEREysrKLl682Llz5+qHnnvuuV69ep0+fTonJ6esrKz6ohhfffVVhw4drr766iapFIZFCwmAiMjIkSNrvKlr/fr1mZmZzz///N///vfg4OCNGzeuX7/e/YTKyspVq1YNHz68iQqFcdFCAiAiEhQUdPLkyW+//db1xMLz58+/9NJLiYmJ8fHx7dq169KlS//+/Xfv3r1r165ffvnluuuua9GiRU5OzqxZs373u9959pFR8E3MsgPwH//4xz/++c9/du/e3WazpaamjhkzZsaMGS1btnSd8PPPP7/yyis7duyw2+3NmjULCgqaPXu269YloCEIJAD/xW63p6amNm/efODAgZe6Z0s7p1mzZoMHD+a+LngKgQQAUML/A9I3FVI1xzGcAAAAAElFTkSuQmCC\n",
       "text/plain": [
        ""
       ]
@@ -433,13 +433,6 @@
    "source": [
     "We can see that this plot of $\\eta$, the $y$ position normalized by the boundary-layer thickness, vs. nondimensional velocity matches the original figure."
    ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {