58.11.21 problem 21

Internal problem ID [14752]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 21
Date solved : Thursday, October 02, 2025 at 09:50:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

ode:=diff(diff(y(x),x),x)+y(x) = x*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-x^{2}+4 c_1 \right ) \cos \left (x \right )}{4}+\frac {\sin \left (x \right ) \left (x +4 c_2 \right )}{4} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to \frac {1}{8} \left (\left (-2 x^2+1+8 c_1\right ) \cos (x)+2 (x+4 c_2) \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.074 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {x^{2}}{4}\right ) \cos {\left (x \right )} + \left (C_{2} + \frac {x}{4}\right ) \sin {\left (x \right )} \]