69.17.35 problem 585

Internal problem ID [18371]
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 : 585
Date solved : Thursday, October 02, 2025 at 03:11:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 40

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

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

✓ Sympy. Time used: 0.221 (sec). Leaf size: 22

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

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