65.14.3 problem 9

Internal problem ID [15824]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.3, page 255
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:28:20 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+9 y&=1 \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: 0.094 (sec). Leaf size: 12

ode:=diff(diff(y(x),x),x)+9*y(x) = 1; 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x),method='laplace');
 

\[ y = -\frac {\cos \left (3 x \right )}{9}+\frac {1}{9} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 17

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

\begin{align*} y(x)&\to \frac {2}{9} \sin ^2\left (\frac {3 x}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \frac {1}{9} - \frac {\cos {\left (3 x \right )}}{9} \]