Answer
See the proof as shown below.
Work Step by Step
Need to verify the differential equation: $t\dfrac{dy}{dt}=y+t^2 \sin t$
$t[-\cos t+ t \sin t-1] =[-t \cos t- t]+t^2 \sin t$
or, $- t \cos t +t^2 \sin t-t=-t \cos t- t+t^2 \sin t$
or, $-t \cos t- t=-t \cos t- t$
Now check the initial conditions.
Since, we have $y=-t \cos t- t$
when $t =\pi$
Then $y(\pi )=-\pi \cos \pi-\pi$
or, $0=0$
Hence, it has been verified .
