Calculate the inverse Laplace transform of $s^2Y(s)+asY(s)+bY(s)$ where $a,b\in\mathbb R$ and $y(0),y'(0)\neq0$

Question

The Laplace Transform of the derivative of a function $y(t)$ with initial value $y(0)$ is $sY(s)-y(0)$ while the Laplace Transform of the second derivative of a function $y(t)$ with initial values $y(0)$ and $y'(0)$ is $s^2Y(s)-sy(0)-y'(0)$.

I would like to calculate the inverse Laplace transform of $s^2Y(s)+asY(s)+bY(s)$ where $a,b\in\mathbb R$ and $y(0),y'(0)\neq0$. If $y(0)=y'(0)=0$ the inverse Laplace transform would be simply $y''(t)+ay'(t)+by(t)$, but i don't know how to proceed in this case where $y(0),y'(0)\neq0$.

Answer 1

Hint: $$sY(s)=\mathcal{L}(y'(t))+y(0)=\mathcal{L}(y'(t)+y(0)\delta(t))$$ $$s^2Y(s)=\mathcal{L}(y''(t))+sy(0)+y'(0)=\mathcal{L}(y''(t)+y(0)\delta'(t)+y'(0)\delta(t))$$

Can u end it now?

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