83.15.6 problem 4 (f)

Internal problem ID [22030]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 4 (f)
Date solved : Thursday, October 02, 2025 at 08:21:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=2 \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&={\frac {3}{2}} \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 26

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

\begin{align*} y(t)&\to \frac {1}{2} e^{-t} \left (-t+2 e^{4 t}+1\right ) \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 17

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

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