Answer
See below
Work Step by Step
Given
$y'''+t^2y'-e^ty=t$
We introduce new variables $x_1$ and $x_2$ defined by
$x_1=y\\
x_2=\frac{dy}{dt}\\
x_3=\frac{d^2y}{dt}$
Then the given differential equation can be replaced by the first-order system
$\frac{dx_3}{dt}+t^2x_2-e^tx_1=t\\
x_2=\frac{dx_1}{dt} \\
x_3=\frac{dx_2}{dt}$
