69.17.15 problem 565

Internal problem ID [18351]
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. Superposition principle. Exercises page 137
Problem number : 565
Date solved : Thursday, October 02, 2025 at 03:10:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y+y^{\prime \prime }&=x \sin \left (x \right )^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

ode:=diff(diff(y(x),x),x)+4*y(x) = x*sin(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-8 x^{2}+128 c_2 +1\right ) \sin \left (2 x \right )}{128}+\frac {\left (-x +32 c_1 \right ) \cos \left (2 x \right )}{32}+\frac {x}{8} \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 72

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

\begin{align*} y(x)&\to \cos (2 x) \int _1^x-\cos (K[1]) K[1] \sin ^3(K[1])dK[1]+\sin (2 x) \int _1^x\frac {1}{2} \cos (2 K[2]) K[2] \sin ^2(K[2])dK[2]+c_1 \cos (2 x)+c_2 \sin (2 x) \end{align*}

✓ Sympy. Time used: 0.778 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = \frac {x}{8} + \left (C_{1} - \frac {x}{32}\right ) \cos {\left (2 x \right )} + \left (C_{2} - \frac {x^{2}}{16}\right ) \sin {\left (2 x \right )} \]