90.9.2 problem 34

Internal problem ID [25190]
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 : 34
Date solved : Thursday, October 02, 2025 at 11:57:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=12 \,{\mathrm e}^{2 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: 0.042 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = 12*exp(2*t); 
ic:=[y(0) = 1, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 23

ode=D[y[t],{t,2}]+3*D[y[t],{t,1}]+2*y[t]==12*Exp[2*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} \left (-3 e^t+e^{4 t}+3\right ) \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*y(t) - 12*exp(2*t) + 4*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)
 

\[ y{\left (t \right )} = \frac {2 e^{2 t}}{3} - \frac {7 \sqrt {2} \sin {\left (\frac {\sqrt {2} t}{2} \right )}}{3} + \frac {\cos {\left (\frac {\sqrt {2} t}{2} \right )}}{3} \]