87.22.21 problem 21

Internal problem ID [23766]
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 : 21
Date solved : Thursday, October 02, 2025 at 09:44:59 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }-4 y&=25 t \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t)-4*y(t) = 25*t*exp(-t); 
ic:=[y(0) = 0, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{4 t}-\frac {{\mathrm e}^{-t} \left (5 t^{2}+2 t +2\right )}{2} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 31

ode=D[y[t],{t,2}]-3*D[y[t],{t,1}]-4*y[t]==25*t*Exp[-t]; 
ic={y[0]==0,Derivative[1][y][0] ==4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{2} e^{-t} \left (-5 t^2-2 t+2 e^{5 t}-2\right ) \end{align*}

✓ Sympy. Time used: 0.195 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-25*t*exp(-t) - 4*y(t) - 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {5 t^{2}}{2} - t - 1\right ) e^{- t} + e^{4 t} \]