61.4.5 problem Problem 2(e)

Internal problem ID [15330]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(e)
Date solved : Thursday, October 02, 2025 at 10:11:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {2}{13}} \\ y^{\prime }\left (0\right )&={\frac {1}{13}} \\ \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+9*y(t) = exp(-2*t); 
ic:=[y(0) = -2/13, D(y)(0) = 1/13]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 132

ode=D[y[t],{t,2}]+9*y[t]==Exp[-2*t]; 
ic={y[0]==-2/13,Derivative[1][y][0] ==1/13}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \cos (3 t) \left (-\int _1^0-\frac {1}{3} e^{-2 K[1]} \sin (3 K[1])dK[1]\right )+\cos (3 t) \int _1^t-\frac {1}{3} e^{-2 K[1]} \sin (3 K[1])dK[1]-\sin (3 t) \int _1^0\frac {1}{3} e^{-2 K[2]} \cos (3 K[2])dK[2]+\sin (3 t) \int _1^t\frac {1}{3} e^{-2 K[2]} \cos (3 K[2])dK[2]+\frac {1}{39} \sin (3 t)-\frac {2}{13} \cos (3 t) \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) + Derivative(y(t), (t, 2)) - exp(-2*t),0) 
ics = {y(0): -2/13, Subs(Derivative(y(t), t), t, 0): 1/13} 
dsolve(ode,func=y(t),ics=ics)
 

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