54.2.325 problem 904

Internal problem ID [12199]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 904
Date solved : Wednesday, October 01, 2025 at 01:08:33 AM
CAS classification : [[_homogeneous, `class D`]]

\begin{align*} y^{\prime }&=\frac {\sin \left (\frac {y}{x}\right ) \left (y+2 x^{3} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )\right )}{2 \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right )} \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 54

ode:=diff(y(x),x) = 1/2*sin(y(x)/x)*(y(x)+2*x^3*cos(1/2*y(x)/x)*sin(1/2*y(x)/x))/sin(1/2*y(x)/x)/x/cos(1/2*y(x)/x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arctan \left (\frac {2 c_1}{{\mathrm e}^{-\frac {x^{2}}{2}}+{\mathrm e}^{\frac {x^{2}}{2}} c_1^{2}}, \frac {-c_1^{2}+{\mathrm e}^{-x^{2}}}{{\mathrm e}^{-x^{2}}+c_1^{2}}\right ) x \]

✓ Mathematica. Time used: 0.243 (sec). Leaf size: 62

ode=D[y[x],x] == (Csc[y[x]/(2*x)]*Sec[y[x]/(2*x)]*Sin[y[x]/x]*(2*x^3*Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)] + y[x]))/(2*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x \arccos \left (-\tanh \left (\frac {x^2}{2}+c_1\right )\right )\\ y(x)&\to x \arccos \left (-\tanh \left (\frac {x^2}{2}+c_1\right )\right )\\ y(x)&\to 0\\ y(x)&\to -\pi x\\ y(x)&\to \pi x \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**3*sin(y(x)/x) + y(x))/x cannot be solved by the factorable group method