Updated readme

README.md

@@ -58,7 +58,7 @@ 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-');
 title('Analytical vs. Euler')
@@ -104,7 +104,7 @@ 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-');
 title('Analytical vs. Euler')
@@ -128,4 +128,58 @@ saveas(b, 'heun4.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
+%Part b.
+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');
+```
+![Height of Plane vs. Distance](./problem5.png)
+