Write a program $($or use a spreadsheet$)$ to solve:

$y^{\text{'}}=y(e^{-t}-1)$

$y(0)=1$ for $0t20$

using the following schemes:

Euler

Backward Euler

Trapezoidal Method

Second Order Runge$-$Kutta

a$)$ Try the following three values of $t$:$0\mathrm{.}2,1\mathrm{.}9,2\mathrm{.}5$

b$)$ Compare $($graphically$)$ the results with the exact solution. Discuss in terms of the

stability and accuracy information that you know about these schemes.

c$)$ Why were schemes $(2)$ and $(3)$ relatively easy to implement for this problem?