44.25.3 problem 3(c)

Internal problem ID [9461]
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 : Tuesday, September 30, 2025 at 06:19:03 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*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)-y(t) = exp(-t)*t; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 51

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.174 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*exp(-t) - y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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