About Second Shifting Theorem

Question

Given equation:

$$y''+y=h(t)$$

initial conditions, $y(0)=1$ and $y'(0)=0$

where h(t+2$\pi$)=h(t)=$1$, if 0$\leq$t$\leq$$\pi$ or $0$, if $\pi$< t< 2$\pi$

Should I use Laplace Transform t-shifting to solve this question?

Answer 1

You have to use the Laplace transform formula for periiodic functionsLaplace transform formula for periiodic functions.

Your function is: $$f(t)=u(t)-u(t-\pi)$$ Take the Laplace Transform of $f(t)$: $$F(s)=\dfrac 1s- \dfrac {e^{-\pi s}} s$$ So that Laplace transform of $h(t)$ is: $$H(s)=\left (\dfrac 1s- \dfrac {e^{-\pi s}} s \right)\dfrac {1}{1-e^{-2\pi s}}$$ $$H(s)=\dfrac 1 s\left ( \dfrac {1-e^{-\pi s}} {1-e^{-2\pi s}} \right)$$ The original equation: $$y''+y=h(t)$$ Becomes: $$Y(s)(s^2+1)-s=\dfrac 1 s\left ( \dfrac {1-e^{-\pi s}} {1-e^{-2\pi s}} \right)$$