# 7.5E: Constant Coefficient Equations with Piecewise Continuous Forcing Functions (Exercises)

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## Q7.5.1

In Exercises 7.5.1-7.5.20 use the Laplace transform to solve the initial value problem. Graph the solution for Exercise 7.5.6, 7.5.9, 7.5.13, and 7.5.19.

1. $$y''+y=\left\{\begin{array}{cl} 3,& 0\le t<\pi,\$4pt] 0,&t\ge\pi,\end{array}\right. \qquad y(0)=0, \quad y'(0)=0$$ 2. $$y''+y=\left\{\begin{array}{cl} 3,&0\le t<4,\\2t-5,&t > 4,\end{array}\right.\qquad y(0)=1,\quad y'(0)=0$$ 3. $$y''-2y'= \left\{\begin{array}{cl} 4,&0\le t<1,\\[4pt] 6,&t\ge 1,\end{array}\right.\qquad y(0)=-6,\quad y'(0)=1$$ 4. $$y''-y=\left\{\begin{array}{cl} e^{2t},&0\le t< 2,\\[4pt] 1,&t\ge 2,\end{array}\right.\qquad y(0)=3,\quad y'(0)=-1$$ 5. $$y''-3y'+2y= \left\{\begin{array}{rl} 0,&0\le t<1,\\[4pt] 1,&1\le t<2,\\[4pt]-1,&t\ge 2, \end{array}\right.\qquad y(0)=-3,\quad y'(0)=1$$ 6. $$y''+4y= \left\{\begin{array}{cl}|\sin t|,&0\le t<2\pi,\\[4pt] 0,&t\ge 2\pi,\end{array}\right.\qquad y(0)=-3,\quad y'(0)=1$$ 7. $$y''-5y'+4y= \left\{\begin{array}{rl} 1,&0\le t<1\\[4pt] -1,&1\le t<2,\\[4pt] 0,&t\ge 2,\end{array}\right.\qquad y(0)=3,\quad y'(0)=-5$$ 8. $$y''+9y=\left\{\begin{array}{ll}{\cos t,}&{0\leq t<\frac{3\pi }{2},}\\{\sin t,}&{t\geq \frac{3\pi }{2},} \end{array} \right. \quad y(0)=0,\: y'(0)=0$$ 9. $$y''+4y=\left\{\begin{array}{ll}{t,}&{0\leq t<\frac{\pi }{2},}\\{\pi ,}&{t\geq \frac{\pi }{2},} \end{array} \right. \quad y(0)=0,\: y'(0)=0$$ 10. $$y''+y=\left\{\begin{array}{cl}\phantom{-}t,&0\le t<\pi, \\[4pt]-t,&t\ge\pi ,\end{array}\right.\; y(0)=0,\; y'(0)=0$$ 11. $$y''-3y'+2y=\left\{\begin{array}{cl} 0,&0\le t<2,\\2t-4,&t\ge 2,\end{array}\right. ,\quad y(0)=0,\quad y'(0)=0$$ 12. $$y''+y=\left\{\begin{array}{cl} t,&0\le t<2\pi,\\-2t,&t\ge 2\pi,\end{array}\right.\quad y(0)=1,\quad y'(0)=2$$ 13. $$y''+3y'+2y=\left\{\begin{array}{cl}\phantom{-}1,&0\le t<2,\\-1,&t\ge 2,\end{array}\right.\; y(0)=0,\; y'(0)=0$$ 14. $$y''-4y'+3y=\left\{\begin{array}{cl}-1,&0\le t<1,\\\phantom{-}1,&t\ge 1,\end{array}\right.\; y(0)=0,\; y'(0)=0$$ 15. $$y''+2y'+y=\left\{\begin{array}{cl} e^t,&0\le t<1,\\e^t-1,&t\ge 1,\end{array}\right.\; y(0)=3,\; y'(0)=-1$$ 16. $$y''+2y'+y=\left\{\begin{array}{cl} 4e^t,&0\le t<1,\\0,&t\ge 1,\end{array}\right.\; y(0)=0,\; y'(0)=0$$ 17. $$y''+3y'+2y=\left\{\begin{array}{cl} e^{-t},&0\le t<1,\\0,&t\ge 1,\end{array}\right.\; y(0)=1,\; y'(0)=-1$$ 18. $$y''-4y'+4y=\left\{\begin{array}{rl} e^{2t},&0\le t<2,\\-e^{2t},&t\ge 2,\end{array}\right.\; y(0)=0,\; y'(0)=-1$$ 19. $$y''=\left\{\begin{array}{cl}t^2,&0\le t<1,\\-t,&1\le t<2,\\t+1,&t\ge 2,\end{array}\right.\; y(0)=1,\; y'(0)=0$$ 20. $$y''+2y'+2y=\left\{\begin{array}{rl}1,&0\le t<2\pi,\\t,&2\pi\le t<3\pi,\\-1,&t\ge 3\pi,\end{array}\right.\; y(0)=2,\quad y'(0)=-1$$ ## Q7.5.2 21. Solve the initial value problem\[y''=f(t), \quad y(0)=0,\quad y'(0)=0,\nonumber$where$f(t)=m+1,\quad m\le t<m+1,\quad m=0,1,2,\dots.\nonumber$

22. Solve the given initial value problem and find a formula that does not involve step functions and represents $$y$$ on each interval of continuity of $$f$$.

1. $$y''+y=f(t), \quad y(0)=0,\quad y'(0)=0$$;
$$f(t)=m+1,\quad m\pi\le t<(m+1)\pi,\quad m=0,1,2,\dots$$.
2. $$y''+y=f(t), \quad y(0)=0,\quad y'(0)=0$$;
$$f(t)=(m+1)t, \quad 2m\pi\le t<2(m+1)\pi,\quad m=0,1,2,\dots$$ HINT: You'll need the formula $1+2+\cdots+m={m(m+1)\over2}.\nonumber$
3. $$y''+y=f(t), \quad y(0)=0,\quad y'(0)=0$$;
$$f(t)=(-1)^m,\quad m\pi\le t<(m+1)\pi,\quad m=0,1,2,\dots.$$
4. $$y''-y=f(t), \quad y(0)=0,\quad y'(0)=0$$;
$$f(t)=m+1,\quad m\le t<(m+1),\quad m=0,1,2,\dots.$$
HINT: You will need the formula $1+r+...+r^{m}=\frac{1-r^{m+1}}{1-r}(r\neq 1).\nonumber$
5. $$y''+2y'+2y=f(t), \quad y(0)=0,\quad y'(0)=0$$;
$$f(t)=(m+1)(\sin t+2\cos t),\quad 2m\pi\le t<2(m+1)\pi,\quad m=0,1,2,\dots.$$
(See the hint in d.)
6. $$y''-3y'+2y=f(t), \quad y(0)=0,\quad y'(0)=0$$;
7. $$f(t)=m+1,\quad m\le t<m+1,\quad m=0,1,2,\dots.$$
(See the hints in b and d.)

This page titled 7.5E: Constant Coefficient Equations with Piecewise Continuous Forcing Functions (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.