README.md

# 06_initial_value_ode
#Problem 2
a) Analytically
 ![Solution](./prob2analytical.PNG)
b) Eueler's Method
```
y_i=1;%initial conditions
n=3;%number of steps
h=1;%step size
y=zeros(1,n+1);
y(1)=y_i;
for i=1:n;
    y(i+1)=y(i)-y(i)*h;
end
```
c)Heun's Approach
```
y_i=1;%initial conditions
n=3;%number of steps
h=1;%step size
y=zeros(1,n+1);
y(1)=y_i;
for i=1:n;
    y_o=y(i)-y(i)*h;
    y(i+1)=y(i)+((-y(i)+y_o)/2)*h;
end
```
| Time(s) | yTrue | yEuler | yHeuns |
| --- | --- | --- | --- |
| 0 | 1 | 1 | 1 |
| 1 | 0.3679 | 0 | 0.5 |
| 2 | 0.1353 | 0 | 0.25 |
| 3 | 0.0498 | 0 | 0.125 |

#Problem 3
a)
 ![Solution](./prob3analytical.PNG)
b)
```
y_i=1;%initial conditions
dy_i=0;
n=3;%number of steps
h=1;%step size
y=zeros(1,n+1);
y(1)=y_i;
for i=1:n;
    y(i+1)=y(i)-3*(sin(3*(i-1)))*h;
end
```
c)
```
y_i=1;%initial conditions
n=3;%number of steps
h=1;%step size
y=zeros(1,n+1);
y(1)=y_i;
for i=1:n;
    y(i+1)=y(i)+((-3*sin(3*(i-1))+-3*sin(3*(i))))/2*h;
end
```

| Time(s) | yTrue | yEuler | yHeuns |
| --- | --- | --- | --- |
| 0 | 1 | 1 | 1 |
| 1 | -0.99 | 1 | 0.7883 |
| 2 | 0.96 | 0.5766 | 0.9958 |
| 3 | -0.91 | 1.4149 | 0.7967 |

#Problem 4
a)
 ![Solution](./prob4analytical.PNG)
```
g=9.81;
cd=.25;
m=60;
for t=1:13
    v(t)=sqrt((g*m)/cd)*tanh(sqrt((g*cd)/m)*(t-1));
    x(t)=1000-(m/cd)*log(cosh(sqrt((g*cd)/m)*(t-1)));
end
```
b)
```
g=9.81;
cd=.25;
m=60;
x_i=1000;
v_i=0;
timesteps=12;
h=1;
v=zeros(1,timesteps+1);
x=zeros(1,timesteps+1);
v(1)=v_i;
x(1)=x_i;
for i=1:timesteps
    v(i+1)=v(i)+(g-(cd/m)*v(i)^2)*h;
    x(i+1)=x(i)-v(i)*h;
end
```
c)
```
g=9.81;
cd=.25;
m=60;
x_i=1000;
v_i=0;
timesteps=12;
h=1;
v=zeros(1,timesteps+1);
x=zeros(1,timesteps+1);
v(1)=v_i;
x(1)=x_i;
for i=1:timesteps
    v_o=v(i)+(g-(cd/m)*v(i)^2)*h;
    v(i+1)=v(i)+(((g-(cd/m)*v(i)^2)+v_o)/2)*h;
    x_o=-v(i)*h;
    x(i+1)=x(i)+(-v(i)+x_o)/2*h;
end
```

| Time (s)| yTrue (m)| yEuler (m) | yHeuns (m) |
| --- | --- | --- | --- |
| 0 | 1000 | 1000  | 1000  |
| 1 | 995.13 | 1000  | 1000  |
| 2 | 980.8924 | 990.19 | 990.190  |
| 3 | 958.323 |  970.97 | 966.066 |
| 4 | 928.8272 | 943.48  | 922.495 |
| 5 | 893.898 | 909.33   | 855.238 |
| 6 | 854.897 | 870.23 | 763.391  |
| 7 | 812.945 | 827.69 | 650.960  |
| 8 | 768.911 | 782.88 | 525.173  |
| 9 | 723.4515 | 736.62 | 392.609  |
| 10 |  671.962 | 689.47  | 257.175  |
| 11 | 629.82 | 641.78 | 120.640  |
| 12 | 582.22 | 593.75 | -16.298  |

| Time Step | vTrue | vEuler | vHuens |
| --- | --- | --- | --- |
| 0 | 0 | 0 | 0 |
| 1 | 9.68 | 9.81 |  9.81 |
| 2 | 18.61 | 19.21902 | 24.12402 |
| 3 | 26.28 |  27.48997 | 43.57116 |
| 4 | 32.45 | 34.15123  | 67.25655  |
| 5 | 37.17 | 39.10162   | 91.84714   |
| 6 | 40.64 | 42.54105 | 112.43114 |
| 7 | 43.11 | 44.81046 | 125.78687 |
| 8 | 44.84 | 46.25389 | 132.56390 |
| 9 | 46.07 | 47.14963 | 135.43424 |
| 10 | 46.84 | 47.69676  | 136.53455  |
| 11 | 47.59 | 48.02768 | 136.93815  |
| 12 | 47.77 | 48.22660 | 137.08365 |

#Problem 5
```
g=9.8;
v_t=5.5;
Cd=1/40;
Cl=1;
v_i=10;
theta_i=0;
y_i=2;
x_i=0;
t=20;
h1=0.1;
h2=0.01;
dt1=t/h1;
dt2=t/h2;
v1=zeros(1,dt1+1);
v1(1)=v_i;
theta1=zeros(1,dt1+1);
theta1(1)=theta_i;
y1=zeros(1,dt1+1);
y1(1)=y_i;
x1=zeros(1,dt1+1);
x1(1)=x_i;
v2=zeros(1,dt2+1);
v2(1)=v_i;
theta2=zeros(1,dt2+1);
theta2(1)=theta_i;
y2=zeros(1,dt2+1);
y2(1)=y_i;
x2=zeros(1,dt2+1);
x2(1)=x_i;
for i=1:dt1
    v1(i+1)=v1(i)+h1*(-g*sin(theta1(i))-(Cd/Cl)*(g/v_t^2)*(v1(i)^2));
    theta1(i+1)=theta1(i)+h1*(-(g/v1(i))*cos(theta1(i))+(g/v_t^2)*v1(i));
    x1(i+1)=x1(i)+h1*v1(i)*cos(theta1(i));
    y1(i+1)=y1(i)+h1*v1(i)*sin(theta1(i));
end
for i=1:dt2
    v2(i+1)=v2(i)+h2*(-g*sin(theta2(i))-(Cd/Cl)*(g/v_t^2)*(v2(i)^2));
    theta2(i+1)=theta2(i)+h2*(-(g/v2(i))*cos(theta2(i))+(g/v_t^2)*v2(i));
    x2(i+1)=x2(i)+h2*v2(i)*cos(theta2(i));
    y2(i+1)=y2(i)+h2*v2(i)*sin(theta2(i));
end
plot(x1,y1,'l',x2,y2)
xlabel('x (m)')
ylabel('y (m)')
```

 ![Plane Flight](./planeplot.png)