How can I solve the ODE $2y'' + ty' -2y=10, y(0) =y’(0)=0 $ by Laplace transform?

Question

Help me to solve this ODE by Laplace Transform:

$$2y''(t) + ty'(t) -2y(t)=10, y(0) =y’(0)=0 $$ Answer: $$ y(t) = 5t^2 /2 $$ How to make the Laplace transform in $$t y’(t)$$? , is $$-[s y(s)]’ $$?

Answer 1

You need to use this : $$\mathcal{L}({ty'})=-\frac {d \mathcal {L}(y')}{ds}=-\frac {d( sY(s)-y(0))}{ds}=-(Y(s)+sY'(s))$$