Solve t^2y''+ty'-y=2t^2 sin t t not equal to zero
Use method of variation of parameters
Solve ∫t^2 sin t dt= -t^2 cos t + 2t sin t +2 cos t
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The solution is attached below in two formats. one is in Word XP Format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
I assume we could use the integral you provided as is.
One must first solve the associated homogenous equation
This equation can be solved using a "guess":
An ODE and an integral are solved. The solution is detailed and well presented.