Internal problem ID [12341]
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(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=1-\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 4.985 (sec). Leaf size: 38
dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=1-Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}-\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1}-{\mathrm e}^{-t}+\frac {{\mathrm e}^{-2 t}}{2}+\frac {1}{2} \]
✓ Solution by Mathematica
Time used: 0.099 (sec). Leaf size: 36
DSolve[{y''[t]+3*y'[t]+2*y[t]==1-DiracDelta[t-1],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{2} e^{-2 t} \left (\left (e^t-1\right )^2-2 e \left (e^t-e\right ) \theta (t-1)\right ) \]