Internal problem ID [12338]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page
368
Problem number: Problem 5(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 \pi ^{2} y=3 \delta \left (t -\frac {1}{3}\right )-\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 5.843 (sec). Leaf size: 36
dsolve([diff(y(t),t$2)+(2*Pi)^2*y(t)=3*Dirac(t-1/3)-Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (-3 \sqrt {3}\, \cos \left (2 \pi t \right )-3 \sin \left (2 \pi t \right )\right ) \operatorname {Heaviside}\left (t -\frac {1}{3}\right )-2 \sin \left (2 \pi t \right ) \operatorname {Heaviside}\left (t -1\right )}{4 \pi } \]
✓ Solution by Mathematica
Time used: 0.125 (sec). Leaf size: 49
DSolve[{y''[t]+(2*Pi)^2*y[t]==3*DiracDelta[t-1/3]-DiracDelta[t-1],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to -\frac {2 \theta (t-1) \sin (2 \pi t)+3 \theta (3 t-1) \left (\sin (2 \pi t)+\sqrt {3} \cos (2 \pi t)\right )}{4 \pi } \]