44.11.10 problem 2(d)

Internal problem ID [9280]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.3. THE METHOD OF VARIATION OF PARAMETERS. Page 71
Problem number : 2(d)
Date solved : Tuesday, September 30, 2025 at 06:16:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

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

\[ y = \frac {\left (x^{2}+4 c_2 -1\right ) \sin \left (x \right )}{4}+\frac {\cos \left (x \right ) \left (x +4 c_1 \right )}{4} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 54

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

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

✓ Sympy. Time used: 0.080 (sec). Leaf size: 20

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

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