Internal problem ID [2369]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.7.
page 704
Problem number: Problem 38.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y+10 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (t +\frac {\pi }{4}\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 67
dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=10*Heaviside(t-Pi/4)*sin(t-Pi/4),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = -2 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) {\mathrm e}^{-2 t +\frac {\pi }{2}}+5 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) {\mathrm e}^{-t +\frac {\pi }{4}}-2 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{2}\right ) \sqrt {2}\, \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right )-{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.065 (sec). Leaf size: 76
DSolve[{y''[t]+3*y'[t]+2*y[t]==10*UnitStep[t-Pi/4]*Sin[t-Pi/4],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} e^{-2 t} \left (-1+2 e^t\right ) & 4 t\leq \pi \\ -\sqrt {2} (2 \cos (t)+\sin (t))-e^{-2 t} \left (1+2 e^{\pi /2}\right )+e^{-t} \left (2+5 e^{\pi /4}\right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}