1. Consider
   
   $$\xot x + x = cos(t) \quad x(0) = 1.$$
   
   
   1. Determine the exact solution.
   2. Using Euler's method, write a computer program in any language other than a matlab to determine an approximate solution to the equation.
   3. Using the fourth order Runge-Kutta method, write a computer program in any language other than matlab to determine an approximate solution to the equation.
   4. In addition to submitting your work for the exact solution and a listing of each program, submit the a figure plotting the exact solution, the solution from Euler's method and the solution from the fourth order Runge-Kutta method on the same plot from $t=0$ to $t=10$ and on a separate plot for t=0 to t=0.5 for:
      1. $h = 1.0$
      2. $h = 0.5$
      3. $h = 0.2$
      4. $h = 0.1$

2. Consider
   
   $$\xot x + x = cos(t) + 5000 cos(100 \pi t) \quad x(0) = 1.$$
   
   
   1. Using Euler's method, write a computer program in any language other than a matlab to determine an approximate solution to the equation.
   2. Using the fourth order Runge-Kutta method, write a computer program in any language other than matlab to determine an approximate solution to the equation.
   3. In addition to submitting your work for the exact solution and a listing of each program, submit the following a figure plotting the exact solution, the solution from Euler's method and the solution from the fourth order Runge-Kutta method on the same plot from t=0 to t=10 for:
      1. $h = 1.0$
      2. $h = 0.1$
      3. $h = 0.01$
      4. $h = 0.001$
   4. For the same time steps, submit four plots where the time range is from $t=0$ to $t=0.5$.
   5. Explain what happens between $h = 0.1$ and $h = 0.01$.