Internal problem ID [859]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 y^{\prime }+3 y-\sin \relax (t )-\left (\delta \left (-3 \pi +t \right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 54
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+3*y(t)=sin(t)+Dirac(t-3*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \frac {{\mathrm e}^{-t} \cos \left (t \sqrt {2}\right )}{4}+\frac {\sqrt {2}\, {\mathrm e}^{3 \pi -t} \sin \left (\sqrt {2}\, \left (-3 \pi +t \right )\right ) \theta \left (-3 \pi +t \right )}{2}+\frac {\sin \relax (t )}{4}-\frac {\cos \relax (t )}{4} \]
✓ Solution by Mathematica
Time used: 0.772 (sec). Leaf size: 80
DSolve[{y''[t]+2*y'[t]+3*y[t]==Sin[t]+DiracDelta[t-3*Pi],{y[0]==0,y'[0]==1/2}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{4} e^{-t} \left (-2 \sqrt {2} e^{3 \pi } \theta (t-3 \pi ) \sin \left (\sqrt {2} (3 \pi -t)\right )+\sqrt {2} \sin \left (\sqrt {2} t\right )+\cos \left (\sqrt {2} t\right )+e^t (\sin (t)-\cos (t))\right ) \\ \end{align*}