75.17.7 problem 557

Internal problem ID [16977]
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 : 557
Date solved : Thursday, March 13, 2025 at 09:04:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 27

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

\[ y = \sin \left (2 x \right ) c_{2} +\cos \left (2 x \right ) c_{1} -\frac {2 \cos \left (x \right )}{15}+\frac {2 \cos \left (x \right )^{3}}{5} \]

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 68

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

\[ y(x)\to \sin (2 x) \int _1^x\frac {1}{4} \sin (K[2]) \sin (4 K[2])dK[2]+\cos (2 x) \int _1^x-2 \cos ^2(K[1]) \sin ^3(K[1])dK[1]+c_1 \cos (2 x)+c_2 \sin (2 x) \]

✓ Sympy. Time used: 0.715 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = \left (C_{1} + \frac {\sin {\left (3 x \right )}}{24} - \frac {\sin {\left (5 x \right )}}{40}\right ) \sin {\left (2 x \right )} + \left (C_{2} - \frac {2 \cos ^{5}{\left (x \right )}}{5} + \frac {2 \cos ^{3}{\left (x \right )}}{3}\right ) \cos {\left (2 x \right )} \]