75.25.5 problem 761

Internal problem ID [17157]
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 18.3. Finding periodic solutions of linear differential equations. Exercises page 187
Problem number : 761
Date solved : Monday, March 31, 2025 at 03:43:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\sin \left (x \right )^{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 31

ode:=diff(diff(y(x),x),x)+9*y(x) = sin(x)^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x +24 c_1 \right ) \cos \left (3 x \right )}{24}+\frac {\left (144 c_2 -1\right ) \sin \left (3 x \right )}{144}+\frac {3 \sin \left (x \right )}{32} \]

✓ Mathematica. Time used: 0.475 (sec). Leaf size: 76

ode=D[y[x],{x,2}]+9*y[x]==Sin[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.290 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*y(x) - sin(x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} \sin {\left (3 x \right )} + \left (C_{1} + \frac {x}{24}\right ) \cos {\left (3 x \right )} + \frac {3 \sin {\left (x \right )}}{32} \]