69.16.56 problem 529

Internal problem ID [18316]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Trial and error method. Exercises page 132
Problem number : 529
Date solved : Thursday, October 02, 2025 at 03:10:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 54

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

\begin{align*} y(x)&\to \sin (x) \int _1^x4 \cos ^2(K[2]) K[2]dK[2]+\cos (x) \int _1^x-2 K[1] \sin (2 K[1])dK[1]+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 17

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

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