Initial Value Problem with Laplace Transform

Question

How do you solve the following with Laplace Transform?

$$ {\rm y}''\left(t\right) - 10\,{\rm y}'\left(t\right) + 25\,{\rm y}\left(t\right) = 24\,t\,{\rm e}^{-2t}\,; \qquad\qquad {\rm y}\left(0\right) = -2\,,\quad {\rm y}'\left(0\right) = -10 $$

Answer 1

Hints:

• $\mathcal{L} (y''(t)) = s^2y(s) -s y(0) -y'(0)$
• $\mathcal{L} (-10y'(t)) = -10(sy(s) -y(0))$
• $\mathcal{L} (25y(t)) = 25y(s)$
• $\mathcal{L} (24te^{-2t}) = \dfrac{24}{(s + 2)^2}$

Now, substitute the ICs, isolate $y(s)$ on the LHS, everything else on the RHS, do a partial fraction expansion and then find the Inverse Laplace Transform. You should end up with the following.

Spoiler

$y(t)=\dfrac{2}{343} e^{-2t} (84 t+e^{7 t} (84 t-367)+24)$