82.4.13 problem 28-13

Internal problem ID [21831]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-13
Date solved : Thursday, October 02, 2025 at 08:02:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t)+2*y(t) = exp(-t); 
ic:=[y(1) = 0, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-t}}{6}-\frac {{\mathrm e}^{-2+t}}{2}+\frac {{\mathrm e}^{-3+2 t}}{3} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 32

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

\begin{align*} y(t)&\to \frac {1}{6} e^{-t-3} \left (e-e^t\right )^2 \left (2 e^t+e\right ) \end{align*}

✓ Sympy. Time used: 0.071 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = \left (\frac {t}{4} - \frac {1}{8}\right ) e^{- t} - \frac {e^{t}}{8 e^{2}} \]