Internal problem ID [2878]
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]]
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=-10 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (t +\frac {\pi }{4}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 2.375 (sec). Leaf size: 63
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)
\[ y \left (t \right ) = -2 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) {\mathrm e}^{\frac {\pi }{2}-2 t}+5 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) {\mathrm e}^{-t +\frac {\pi }{4}}-2 \sqrt {2}\, \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{2}\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right )-{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.143 (sec). Leaf size: 87
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]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-2 t} \left (-1+2 e^t\right ) & 4 t\leq \pi \\ -e^{-2 t} \left (2 \sqrt {2} e^{2 t} \cos (t)-2 e^t-5 e^{t+\frac {\pi }{4}}+\sqrt {2} e^{2 t} \sin (t)+2 e^{\pi /2}+1\right ) & \text {True} \\ \end {array} \\ \end {array} \]