87.22.34 problem 34

Internal problem ID [23779]
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 : 34
Date solved : Thursday, October 02, 2025 at 09:45:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=2 t^{2}+2 \,{\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: 0.046 (sec). Leaf size: 24

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

\[ y = -4-2 t^{2}+3 \,{\mathrm e}^{t}-{\mathrm e}^{-t} \left (t -1\right ) \]

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 28

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

\begin{align*} y(t)&\to -2 \left (t^2+2\right )-e^{-t} (t-1)+3 e^t \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = - 2 t^{2} + \left (1 - t\right ) e^{- t} + 3 e^{t} - 4 \]