- Assume a solution of the form

*Hint:*you could use a power series for the homogeneous solution and then undetermined coefficients for the particular solution, but since the inhomogeneous term is*t*you might was well just include it when you determine the coefficients in the power series. - Write a computer program that uses Euler's method to compute an approximate numerical solution for the equation in the previous problem. On the same plot, plot the numerical solution and the power series solutions.
- Assume a solution of the form
- Write a computer program that uses Euler's method to compute an approximate numerical solution for the equation in the previous problem. On the same plot, plot the numerical solution and the power series solutions.
- Assume a solution of the form
*Hint:*expand the sine and exponential functions in their Taylor series. - Assume a solution of the form

*Hint:*you want to equate powers of (t-1), not t. - Write a computer program that uses Euler's method to compute an approximate numerical solution for the equation in the previous problem. On the same plot, plot the numerical solution and the power series solutions.
*Extra credit:*(5 points each) for each of the problems determine a general form for the series (with a summation sign and an index on the coefficients).*Extra credit:*(5 points each) if you did the previous extra credit for each one, use your answers there to compute the range of times for which the power series will converge.*Extra credit:*(5 points each) for the problems where initial conditions were specified, plot the series solutions for 25, 50, 75 and 100 terms and compare with the approximate numerical solutions.

**Notes:**

- I have not done the extra credit problems myself (yet). Sometimes they can be pretty hard to work out; other times, not so much. At this point I don't know which case it is for any of these problems.
- The usual rules for programs apply: you must use Fortran, C or C++ for the program to determine approximate numerical solutions using Euler's method. You may use Matlab for all the plots.
- If the problem asks you to plot four and then five terms in the series, if one of the coefficients happens to be zero so that you are plotting exactly the same thing, then you must keep including more terms until you plot the next higher non-zero term in the series. The point is to add a term to see how it changes, which should indicate the range in which the truncated series is a good approximation. This shouldn't be confused with the range of convergence of the whole (infinite) series.