Internal problem ID [13575]
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 (d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _quadrature]]
\[ \boxed {y^{\prime \prime }=\delta \left (-1+t \right )-\delta \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 22
dsolve([diff(y(t),t$2)=Dirac(t-1)-Dirac(t-4),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \left (-t +4\right ) \operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -1\right ) \left (t -1\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 23
DSolve[{y''[t]==DiracDelta[t-1]-DiracDelta[t-4],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to (t-1) \theta (t-1)-(t-4) \theta (t-4) \]