89.16.42 problem 42

Internal problem ID [24692]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Exercises at page 140
Problem number : 42
Date solved : Thursday, October 02, 2025 at 10:47:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\sin \left (3 x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y \left (\frac {\pi }{2}\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)+9*y(x) = sin(3*x); 
ic:=[y(0) = 1, y(1/2*Pi) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\sin \left (3 x \right )+\cos \left (3 x \right )-\frac {\cos \left (3 x \right ) x}{6} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 23

ode=D[y[x],{x,2}]+9*y[x]== Sin[3*x]; 
ic={y[0]==1,y[Pi/2]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sin (3 x)-\frac {1}{6} (x-6) \cos (3 x) \end{align*}

✓ Sympy. Time used: 0.072 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*y(x) - sin(3*x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, y(pi/2): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (1 - \frac {x}{6}\right ) \cos {\left (3 x \right )} - \sin {\left (3 x \right )} \]