Solve
\begin{equation*}
\left\{ \begin{array}{ll}
x'(t)-x(t)+y(t)=\mathcal{U}(t-2) \\
-x(t)+y'(t)+y(t)=0 \\
x(0)=y(0)=0.
\end{array} \right.
\end{equation*}
Section 4.4 The Laplace transform of linear systems
Some linear systems are easier to solve using the Laplace transform rather than eigenpair techniques. Here we take the Laplace transform of a set of linear differential equations and use algebraic techniques to solve for \(X(s)\) and \(Y(s)\text{.}\) We then use the inverse transform to find the \(x(t)\) and \(y(t)\) that satisfy our system.
Example 4.4.1.
Example 4.4.2.
Solve
\begin{equation*}
\left\{ \begin{array}{ll}
x''(t)+3y'(t)+3y(t)=0 \\
x''(t)+3y(t)=e^{-t} \\
x(0)=x'(0)=y(0)=0
\end{array} \right.
\end{equation*}
