Internal problem ID [13577]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number: 31.6 (f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y=\delta \left (t \right )+\delta \left (t -\pi \right )} \]
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 22
dsolve(diff(y(t),t$2)+y(t)=Dirac(t)+Dirac(t-Pi),y(t), singsol=all)
\[ y = \cos \left (t \right ) y \left (0\right )+\sin \left (t \right ) \left (\operatorname {Heaviside}\left (-t +\pi \right )+y^{\prime }\left (0\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.118 (sec). Leaf size: 92
DSolve[y''[t]+2*y[t]==DiracDelta[t]+DiracDelta[t-Pi],y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to -\frac {\theta (t-\pi ) \sin \left (\sqrt {2} (\pi -t)\right )}{\sqrt {2}}+\frac {\theta (t) \sin \left (\sqrt {2} t\right )}{\sqrt {2}}-\frac {\cos \left (\sqrt {2} \pi \right ) \sin \left (\sqrt {2} t\right )}{\sqrt {2}}+c_1 \cos \left (\sqrt {2} t\right )+c_2 \sin \left (\sqrt {2} t\right ) \]