Answer
See below
Work Step by Step
Given
$\frac{d^2x}{dt^2}-3\frac{dy}{dt}+x=\sin t\\
\frac{d^2y}{dt^2}-t\frac{dx}{dt}+e^ty=t^2$
We introduce new variables:
$x_1=x\\
x_2=\frac{dx}{dt}\\
x_3=y\\
x_4=\frac{dy}{dt}$
Then the given differential equation can be replaced by
$\frac{dx_2}{dt}-3x_4+x_1=\sin t\\
\frac{dx_4}{dt}-tx_2e^tx_3=t^2$
Hence,
$x_2=\frac{dx_1}{dt}\\
\frac{dx_2}{dt}=3x_4-x_1+\sin t\\
\frac{dx_4}{dt}=tx_2+e^tx^3+t^2\\
x_4=\frac{dx_3}{dt}$
