Internal problem ID [858]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+3 y^{\prime }+2 y-\left (\delta \left (t -5\right )\right )-\theta \left (-10+t \right )=0} \end {gather*} With initial conditions \begin {align*} \left [y \relax (0) = 0, y^{\prime }\relax (0) = {\frac {1}{2}}\right ] \end {align*}
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 70
dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=Dirac(t-5)+Heaviside(t-10),y(0) = 0, D(y)(0) = 1/2],y(t), singsol=all)
\[ y \relax (t ) = -\frac {{\mathrm e}^{-2 t}}{2}+\theta \left (t -5\right ) {\mathrm e}^{-t +5}-\theta \left (t -5\right ) {\mathrm e}^{-2 t +10}+\frac {\theta \left (-10+t \right )}{2}-\theta \left (-10+t \right ) {\mathrm e}^{10-t}+\frac {\theta \left (-10+t \right ) {\mathrm e}^{20-2 t}}{2}+\frac {{\mathrm e}^{-t}}{2} \]
✓ Solution by Mathematica
Time used: 0.163 (sec). Leaf size: 71
DSolve[{y''[t]+3*y'[t]+2*y[t]==DiracDelta[t-5]+UnitStep[t-10],{y[0]==0,y'[0]==1/2}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{2} e^{-2 t} \left (2 e^5 \left (e^t-e^5\right ) \theta (t-5)+\left (e^{10}-e^t\right )^2 (-\theta (10-t))+e^t+e^{2 t}-2 e^{t+10}+e^{20}-1\right ) \\ \end{align*}