67.19.10 problem 28.9 (c)

Internal problem ID [16895]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 28. The inverse Laplace transform. Additional Exercises. page 509
Problem number : 28.9 (c)
Date solved : Thursday, October 02, 2025 at 01:40:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-9 y&=24 \,{\mathrm e}^{-3 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=6 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)-9*y(t) = 24*exp(-3*t); 
ic:=[y(0) = 6, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (-4 t +2\right ) {\mathrm e}^{-3 t}+4 \,{\mathrm e}^{3 t} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 23

ode=D[y[t],{t,2}]-9*y[t]==24*Exp[-3*t]; 
ic={y[0]==6,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-3 t} \left (-4 t+4 e^{6 t}+2\right ) \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*y(t) + Derivative(y(t), (t, 2)) - 24*exp(-3*t),0) 
ics = {y(0): 6, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (2 - 4 t\right ) e^{- 3 t} + 4 e^{3 t} \]