90.8.6 problem 6

Internal problem ID [25180]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 2. The Laplace Transform. Exercises at page 109
Problem number : 6
Date solved : Thursday, October 02, 2025 at 11:57:37 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 13

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

\[ y = \frac {{\mathrm e}^{-2 t} t^{2}}{2} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 17

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

\begin{align*} y(t)&\to \frac {1}{2} e^{-2 t} t^2 \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 12

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

\[ y{\left (t \right )} = \frac {t^{2} e^{- 2 t}}{2} \]