Internal problem ID [862]
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: 7.
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 }+2 y-\cos \relax (t )-\left (\delta \left (t -\frac {\pi }{2}\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: 0.024 (sec). Leaf size: 47
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=cos(t)+Dirac(t-Pi/2),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = -\theta \left (t -\frac {\pi }{2}\right ) \cos \relax (t ) {\mathrm e}^{-t +\frac {\pi }{2}}+\frac {\left (-\cos \relax (t )-3 \sin \relax (t )\right ) {\mathrm e}^{-t}}{5}+\frac {\cos \relax (t )}{5}+\frac {2 \sin \relax (t )}{5} \]
✓ Solution by Mathematica
Time used: 0.076 (sec). Leaf size: 50
DSolve[{y''[t]+2*y'[t]+2*y[t]==Cos[t]+DiracDelta[t-Pi/2],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{5} e^{-t} \left (\left (-5 e^{\pi /2} \theta (2 t-\pi )+e^t-1\right ) \cos (t)+\left (2 e^t-3\right ) \sin (t)\right ) \\ \end{align*}