90.9.6 problem 38

Internal problem ID [25194]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 2. The Laplace Transform. Exercises at page 139
Problem number : 38
Date solved : Thursday, October 02, 2025 at 11:57:42 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 (0\right )&=-3 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 15

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

\[ y = -{\mathrm e}^{t} \left (-4 \,{\mathrm e}^{t}+7+t \right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 19

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

\begin{align*} y(t)&\to e^t \left (-t+4 e^t-7\right ) \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = \left (- \frac {t}{4} - \frac {11}{8}\right ) e^{t} - \frac {13 e^{- t}}{8} \]