Math428: Problem Set #2

1.

Write the differential equation

$$y'' + (y')^2/y - sin(y) = 0, \ \ \ \ y(0) = 1, \ \ \ \ y'(0) = 1$$

as a first order system of equations. Note: it will not necessarily be linear. Suggest a general class of $n^{\rm th}$-order nonlinear differential equations that can be converted to a first order system. Try to make your class as broad as possible.

2.

Using Euler's method, solve

$$y' = -y, \ \ \ \ y(0) = 1$$

for $0 \leq t \leq 10$ with step sizes of h=0.5,0.75,1,1.5,2,2.5. Plot the errors in the numerical computation as a function of time for the different step sizes all on the same graph. (Make sure your plot has a legend.) How does the computed solution compare to the exact solution for different values of h? Would a Taylor series method improve this computation at all?

3.

Repeat problem #2 using the backward Euler's method. Explain the difference in behavior by examining the numerical method as a difference equation.

4.

Repeat problem #2 for the differential equation

$$y' = -\left[1+\frac{9}{10}\cos(t)\right] y, \ \ \ \ y(0) = 1$$

for $0 \leq t \leq 20$. If you cannot find an exact solution, compute a reference solution by setting h to be very, very small. Explain the difference in behavior for different values of h by examining the varying decay rate of the exact system as a function of time.