# Section 6.4 – Differential Equations with Discontinuous Forcing Functions

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Solve the following initial value problem using the Laplace transform: $y''+2y'+5y=\sin (t)+u_{\pi}(t)\cos(t-\pi),\qquad y(0)=0, \quad y'(0)=0.$

$$y(t)=h(t)+u_{\pi}(t)p(t-\pi)$$, where $$h(t)=-\dfrac{1}{10}\cos(t)+\dfrac{2}{10}\sin(t)+e^{-t}[\dfrac{1}{10}\cos(2t)-\dfrac{1}{20}\sin(2t)]$$ and $$p(t-\pi)=-\dfrac{2}{10}\cos(t)-\dfrac{1}{10}\sin(t)+e^{-(t-\pi)}[-\dfrac{2}{10}\cos(2t)-\dfrac{3}{20}\sin(2t)]$$

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