Internal problem ID [5754]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 7. Laplace Transforms. Section 7.5 Problesm for review and discovery. Section A,
Drill exercises. Page 309
Problem number: 3(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 y^{\prime }-y-t \,{\mathrm e}^{-t}=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 39
dsolve([diff(y(t),t$2)+2*diff(y(t),t)-y(t)=t*exp(-t),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
\[ y \relax (t ) = \frac {3 \,{\mathrm e}^{\left (\sqrt {2}-1\right ) t} \sqrt {2}}{8}-\frac {3 \,{\mathrm e}^{-\left (1+\sqrt {2}\right ) t} \sqrt {2}}{8}-\frac {t \,{\mathrm e}^{-t}}{2} \]
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 33
DSolve[{y''[t]+2*y'[t]-y[t]==t*Exp[-t],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{4} e^{-t} \left (3 \sqrt {2} \sinh \left (\sqrt {2} t\right )-2 t\right ) \\ \end{align*}