Using Laplace Transforms, solve $y''-2y'+5y=e^t$

Question

Given function:

$$y''-2y'+5y=e^t$$

where $y(\pi)=2$ and $y'(\pi)=3$

I've never seen $y(\pi)=2$ and $y'(\pi)=3$ before, only $y(0)=x$ could someone help explain how to solve this problem?

Answer 1

Hint.

The initial conditions in Laplace transformation can be understood as generic constants so after transforming we have

$$ s^2Y(s)-2sY(s)+5Y(s) = \frac{1}{s-1}+(s-2)C_1+C_2 $$

where $C_1 = y(0), C_2 = y'(0)$. After anti-transformation, we can use $C_1, C_2$ to determine the sought conditions at $t=\pi$.