78.14.12 problem 4 (d)

Internal problem ID [18277]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 19. The Method of Variation of Parameters. Problems at page 135
Problem number : 4 (d)
Date solved : Monday, March 31, 2025 at 05:24:36 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: 0.038 (sec). Leaf size: 34

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

\[ y(x)\to \frac {1}{8} \left (\left (2 x^2-1+8 c_2\right ) \sin (x)+2 (x+4 c_1) \cos (x)\right ) \]

✓ Sympy. Time used: 0.125 (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)
 

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