90.9.1 problem 33

Internal problem ID [25189]
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 : 33
Date solved : Thursday, October 02, 2025 at 11:57:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=9 \,{\mathrm e}^{2 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = 9*exp(2*t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 21

ode=D[y[t],{t,2}]+2*D[y[t],{t,1}]+y[t]==9*Exp[2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.059 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 9*exp(2*t) + 3*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {9 e^{2 t}}{13} - \frac {18 \sqrt {3} \sin {\left (\frac {\sqrt {3} t}{3} \right )}}{13} - \frac {9 \cos {\left (\frac {\sqrt {3} t}{3} \right )}}{13} \]