87.22.8 problem 8

Internal problem ID [23753]
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 : 8
Date solved : Thursday, October 02, 2025 at 09:44:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 13

ode:=diff(diff(y(t),t),t)-4*y(t) = -3*exp(t); 
ic:=[y(0) = -1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 23

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

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

✓ Sympy. Time used: 0.069 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*y(t) + 3*exp(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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