Chapter 9.7, Problem 25E

To find a general solution to the system

$x^{\prime }\left(t\right)=\left[\begin{array}{cc}0& 1\\ -2& 3\end{array}\right]x\left(t\right)+f\left(t\right)$, where $f\left(t\right)=\left[\begin{array}{c}{e}^t\\ 0\end{array}\right]$

proceed as follows:

a. Find a fundamental solution set for the corresponding homogeneous system.

b. The obvious choice for a particular solution would be a vector function of the form $x_p\left(t\right)=e^{t}a$ however, the homogeneous system has a solution of this form. The next choice would be $x_p\left(t\right)=t{e}^{t}a$. Show that this choice does not work.

c. For systems, multiplying by $t$ is not always sufficient. The proper guess is

$x_p\left(t\right)=t{e}^{t}a+e^{t}b$

Use this guess to find a particular solution of the given system.

d. Use the results of parts (a) and (c) to find a general solution of the given system.