67.19.1 problem 28.6 (a)

Internal problem ID [16886]
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 : 28.6 (a)
Date solved : Thursday, October 02, 2025 at 01:40:07 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-9 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4 \\ y^{\prime }\left (0\right )&=9 \\ \end{align*}

✓ Maple. Time used: 0.098 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)-9*y(t) = 0; 
ic:=[y(0) = 4, D(y)(0) = 9]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {7 \,{\mathrm e}^{3 t}}{2}+\frac {{\mathrm e}^{-3 t}}{2} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 23

ode=D[y[t],{t,2}]-9*y[t]==0; 
ic={y[0]==4,Derivative[1][y][0] ==9}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{2} e^{-3 t} \left (7 e^{6 t}+1\right ) \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 4, Subs(Derivative(y(t), t), t, 0): 9} 
dsolve(ode,func=y(t),ics=ics)
 

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