54.2.381 problem 960

Internal problem ID [12255]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 960
Date solved : Wednesday, October 01, 2025 at 01:23:52 AM
CAS classification : [[_homogeneous, `class D`]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 16

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

\[ y = \frac {\arccos \left ({\mathrm e}^{2 x} c_1 +1\right ) x}{2} \]

✓ Mathematica. Time used: 35.124 (sec). Leaf size: 19

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

\begin{align*} y(x)&\to x \arcsin \left (e^{x+c_1}\right )\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (2*x**2*sin(y(x)/x)**2 - y(x)*sin(y(x)/x) + 2*y(x)*sin(3*y(x)/(2*x))*cos(y(x)/(2*x)))/(x*sin(2*y(x)/x)) cannot be solved by the factorable group method