26.9.8 problem Exercise 22.8, page 240

Internal problem ID [7117]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 22. Variation of Parameters
Problem number : Exercise 22.8, page 240
Date solved : Tuesday, September 30, 2025 at 04:21:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=4 x \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+y(x) = 4*sin(x)*x; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 54

ode=D[y[x],{x,2}]+y[x]==4*x*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sin (x) \int _1^x2 K[2] \sin (2 K[2])dK[2]+\cos (x) \int _1^x-4 K[1] \sin ^2(K[1])dK[1]+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 17

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

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