50.25.3 problem 3(c)

Internal problem ID [8175]
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)
Date solved : Monday, January 27, 2025 at 03:45:14 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }-y&=t \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.669 (sec). Leaf size: 25

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)
 

\[ y = \frac {{\mathrm e}^{-t} \left (3 \sqrt {2}\, \sinh \left (\sqrt {2}\, t \right )-2 t \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 51

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]-y[t]==t*Exp[-t],{y[0]==0,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{8} e^{-t} \left (-4 t-3 \sqrt {2} e^{-\sqrt {2} t}+3 \sqrt {2} e^{\sqrt {2} t}\right ) \]