#README.md#

# 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