87.22.5 problem 5

Internal problem ID [23750]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:44:53 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=-3 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 15

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = 2*t*exp(-t); 
ic:=[y(0) = 3, D(y)(0) = -3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-t} \left (t^{3}+9\right )}{3} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 19

ode=D[y[t],{t,2}]+2*D[y[t],t]+y[t]==2*t*Exp[-t]; 
ic={y[0]==3,Derivative[1][y][0] ==-3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{3} e^{-t} \left (t^3+9\right ) \end{align*}

✓ Sympy. Time used: 0.178 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t*exp(-t) + y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): -3} 
dsolve(ode,func=y(t),ics=ics)
 

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