Updated readme

#README.md#

@@ -0,0 +1,153 @@
+# 06_initial_value_ode
+
+## Problem 2
+
+### Script:
+```matlab
+%Part a.
+y_analytical = @(t) exp(-t);
+
+%Part b.
+dt = 0.01; %step size
+a = figure(1);
+set(0, 'defaultAxesFontsize', 16)
+set(0, 'defaultTextFontsize', 16)
+set(0, 'defaultLineLineWidth', 2)
+t = [0:dt:3]';
+y_n = zeros(size(t));
+y_n(1) = 1; %initial condition
+for i = 2:length(t)
+	dy = ode2(t, y_n(i-1));
+    y_n(i) = y_n(i-1) + dt*dy;
+end
+plot(t, y_analytical(t), 'k-', t, y_n, 'o-');
+title('Analytical vs. Euler')
+legend('Analytical', 'Euler', 'Location', 'Northeast')
+saveas(a, 'euler.png');
+
+%Part c.
+b = figure(2);
+y_heun = heun_sol_order1(@(t,y) ode2(t,y), dt, 1, [0 3]);
+plot(t, y_analytical(t), 'k-', t, y_heun, 'o-');
+title('Analytical vs. Heun')
+legend('Analytical', 'Heun', 'Location', 'Northeast')
+saveas(b, 'heun.png');
+```
+
+### Part b.
+![Analytical vs. Euler](./euler2.png)
+
+### Part c.
+![Analytical vs. Heun](./heun2.png)
+
+## Problem 3
+
+### Script:
+```matlab
+%Part a.
+y_analytical = @(t) cos(3*t); 
+
+%Part b.
+dt = 0.001; %step size
+a = figure(1);
+set(0, 'defaultAxesFontsize', 16)
+set(0, 'defaultTextFontsize', 16)
+set(0, 'defaultLineLineWidth', 2)
+t = [0:dt:3]';
+y_n = zeros(length(t), 2);
+y_n(1, :) = [1 0]; %initial condition
+for i = 2:length(t)
+    dy = ode3(t, y_n(i-1, :));
+	y_n(i, :) = y_n(i-1, :) + dt*dy;
+end
+plot(t, y_analytical(t), 'k-', t, y_n(:,1), 'o-');
+title('Analytical vs. Euler')
+legend('Analytical', 'Euler', 'Location', 'Northeast')
+saveas(a, 'euler3.png');
+
+%Part c.
+b = figure(2);
+dt = 0.001;
+t = [0:dt:3];
+y_heun = heun_sol_order2(@(t,y) ode3(t,y), dt, [1 0], [0 3]);
+plot(t, y_analytical(t), 'k-', t, y_heun(:,1), 'o-');
+title('Analytical vs. Heun')
+legend('Analytical', 'Heun', 'Location', 'Northeast')
+saveas(b, 'heun3.png');
+```
+
+### Part b.
+![Analytical vs. Euler](./euler3.png)
+
+### Part c.
+![Analytical vs. Heun](./heun3.png)
+
+## Problem 4
+
+### Script:
+```matlab
+%part a.
+g = 9.81; %m/s^2
+cd = 0.25; %kg/m
+m = 60; %kg
+v_term = sqrt(m*g/cd);
+x_analytical = @(t) ((v_term^2)/g)*log(abs(cosh((g*t)/v_term))) + 100;
+
+%part b.
+dt = 0.01; %step size
+a = figure(1);
+set(0, 'defaultAxesFontsize', 16)
+set(0, 'defaultTextFontsize', 16)
+set(0, 'defaultLineLineWidth', 2)
+t = [0:dt:12]';
+y_n = zeros(length(t), 2);
+y_n(1, :) = [100 0]; %initial condition
+for i = 2:length(t)
+    dy = ode4(t, y_n(i-1, :));
+	y_n(i, :) = y_n(i-1, :) + dt*dy;
+end
+plot(t, x_analytical(t), 'k-', t, y_n(:,1), 'o-');
+title('Analytical vs. Euler')
+legend('Analytical', 'Euler', 'Location', 'Northeast')
+saveas(a, 'euler4.png');
+
+%Part c.
+b = figure(2);
+dt = 0.01;
+t = [0:dt:12];
+y_heun = heun_sol_order2(@(t,y) ode4(t,y), dt, [100 0], [0 12]);
+plot(t, x_analytical(t), 'k-', t, y_heun(:,1), 'o-');
+title('Analytical vs. Heun')
+legend('Analytical', 'Heun', 'Location', 'Northeast')
+saveas(b, 'heun4.png');
+```
+
+### Part b.
+![Analytical vs. Euler](./euler4.png)
+
+### Part c.
+![Analytical vs. Heun](./heun4.png)
+
+## Problem 5
+
+### Part a.
+```matlab
+function dy = phugoid_ode(t,y)
+    %glider equations describing phugoid path
+    %y = [v, theta, x, y]
+    g = 9.81; %m/s^2
+    vt = 5.5; %m/s
+    cl = 5.2;
+    cd = 1;
+    dy = zeros(size(y));
+    dy(1) = -g*sin(y(2)) - cd/cl*g/vt^2*y(1)^2;
+    dy(2) = -(g/y(1))*cos(y(2)) + g/vt^2*y(1);
+    dy(3) = y(1)*cos(y(2));
+    dy(4) = y(1)*sin(y(2));
+end
+```
+
+### Part b.
+```matlab
+
+

HW6Problem2.m

@@ -14,15 +14,15 @@ for i = 2:length(t)
 	dy = ode2(t, y_n(i-1));
     y_n(i) = y_n(i-1) + dt*dy;
 end
-plot(t, y_analytical(t), 'k-', t, y_n, 'o-');
+plot(t, y_n, 'o', t, y_analytical(t), 'k-');
 title('Analytical vs. Euler')
 legend('Analytical', 'Euler', 'Location', 'Northeast')
 saveas(a, 'euler2.png');
 
 %Part c.
 b = figure(2);
 y_heun = heun_sol_order1(@(t,y) ode2(t,y), dt, 1, [0 3]);
-plot(t, y_analytical(t), 'k-', t, y_heun, 'o-');
+plot(t, y_heun, 'o', t, y_analytical(t), 'k-');
 title('Analytical vs. Heun')
 legend('Analytical', 'Heun', 'Location', 'Northeast')
 saveas(b, 'heun2.png');

HW6Problem3.m

@@ -12,9 +12,9 @@ y_n = zeros(length(t), 2);
 y_n(1, :) = [1 0]; %initial condition
 for i = 2:length(t)
     dy = ode3(t, y_n(i-1, :));
-	y_n(i, :) = y_n(i-1, :) + dt(k)*dy;
+	y_n(i, :) = y_n(i-1, :) + dt*dy;
 end
-plot(t, y_analytical(t), 'k-', t, y_n(:,1), 'o-');
+plot(t, y_n(:,1), 'o', t, y_analytical(t), 'k-');
 title('Analytical vs. Euler')
 legend('Analytical', 'Euler', 'Location', 'Northeast')
 saveas(a, 'euler3.png');
@@ -24,7 +24,7 @@ b = figure(2);
 dt = 0.001;
 t = [0:dt:3];
 y_heun = heun_sol_order2(@(t,y) ode3(t,y), dt, [1 0], [0 3]);
-plot(t, y_analytical(t), 'k-', t, y_heun(:,1), 'o-');
+plot(t, y_heun(:,1), 'o-', t, y_analytical(t), 'k-');
 title('Analytical vs. Heun')
 legend('Analytical', 'Heun', 'Location', 'Northeast')
 saveas(b, 'heun3.png');

HW6Problem4.m

@@ -16,9 +16,9 @@ y_n = zeros(length(t), 2);
 y_n(1, :) = [100 0]; %initial condition
 for i = 2:length(t)
     dy = ode4(t, y_n(i-1, :));
-	y_n(i, :) = y_n(i-1, :) + dt(k)*dy;
+	y_n(i, :) = y_n(i-1, :) + dt*dy;
 end
-plot(t, x_analytical(t), 'k-', t, y_n(:,1), 'o-');
+plot(t, y_n(:,1), 'o-', t, x_analytical(t), 'k-');
 title('Analytical vs. Euler')
 legend('Analytical', 'Euler', 'Location', 'Northeast')
 saveas(a, 'euler4.png');
@@ -28,7 +28,7 @@ b = figure(2);
 dt = 0.01;
 t = [0:dt:12];
 y_heun = heun_sol_order2(@(t,y) ode4(t,y), dt, [100 0], [0 12]);
-plot(t, x_analytical(t), 'k-', t, y_heun(:,1), 'o-');
+plot(t, y_heun(:,1), 'o-', t, x_analytical(t), 'k-');
 title('Analytical vs. Heun')
 legend('Analytical', 'Heun', 'Location', 'Northeast')
 saveas(b, 'heun4.png');

HW6Problem5.m

@@ -1,16 +1,30 @@
 %Part b.
-dy = ode3(0, [1;0]);
-dt = [0.1 0.01];
-figure()
-hold on
-for k = 1:length(dt)
-    t = [0:dt(k):20]';
-    y_n = zeros(length(t), 4);
-    y_n(1, :) = [10 0 0 2]; %initial condition
-    for i = 2:length(t)
-        dy = phugoid_ode(t(i-1), y_n(i-1, :));
-        y_n(i, :) = y_n(i-1, :) + dt(k)*dy;
-    end
-    [t23, y23] = ode23(@(t,y) phugoid_ode(t,y), [0,20], [10 0 0 2]);
-    plot(y_n(:,3), y_n(:, 4), '.', y23(:,3), y23(:,4), 's');
-end
+dt = 0.1;
+t = [0:dt:20]';
+y_n1 = zeros(length(t), 4);
+y_n1(1, :) = [10 0 0 2]; %initial condition
+for i = 2:length(t)
+	dy = phugoid_ode(t(i-1), y_n1(i-1, :));
+	y_n1(i, :) = y_n1(i-1, :) + dt*dy;
+end
+
+dt = 0.01;
+t = [0:dt:20]';
+y_n2 = zeros(length(t), 4);
+y_n2(1, :) = [10 0 0 2]; %initial condition
+for i = 2:length(t)
+	dy = phugoid_ode(t(i-1), y_n2(i-1, :));
+	y_n2(i, :) = y_n2(i-1, :) + dt*dy;
+end
+
+a = figure(1);
+set(0, 'defaultAxesFontsize', 16)
+set(0, 'defaultTextFontsize', 16)
+set(0, 'defaultLineLineWidth', 2)
+[t23, y23] = ode23(@(t,y) phugoid_ode(t,y), [0,20], [10 0 0 2]);
+plot(y_n1(:,3), y_n1(:, 4), '.', y_n2(:,3), y_n2(:, 4), 'o', y23(:,3), y23(:,4), '-');
+title('Height of Plane vs. Distance')
+xlabel('Distance (m)')
+ylabel('Height (m)')
+legend('dt = 0.1', 'dt = 0.01', 'analytical', 'Location', 'Northeast')
+saveas(a, 'problem5.png');