54.2.390 problem 969

Internal problem ID [12264]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 969
Date solved : Wednesday, October 01, 2025 at 01:25:44 AM
CAS classification : [[_homogeneous, `class D`]]

\begin{align*} y^{\prime }&=\frac {y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +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 ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+2 \sin \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x}{2 \cos \left (\frac {y}{x}\right ) \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right ) \left (x +1\right )} \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 60.17 (sec). Leaf size: 29

ode=D[y[x],x] == (Csc[y[x]/(2*x)]*Sec[y[x]/(2*x)]*Sec[y[x]/x]*(x*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 + (x*Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)]*y[x])/2 - (Sin[y[x]/x]*y[x])/2 - (x*Sin[y[x]/x]*y[x])/2 + (Cos[y[x]/(2*x)]*Sin[(3*y[x])/(2*x)]*y[x])/2 + (x*Cos[y[x]/(2*x)]*Sin[(3*y[x])/(2*x)]*y[x])/2))/(x*(1 + x)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x \arcsin \left (\exp \left (\int _1^x\frac {1}{K[1]^2+K[1]}dK[1]+c_1\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x*y(x)*sin(y(x)/(2*x))*cos(y(x)/(2*x)) - x*y(x)*sin(y(x)/x) + x*y(x)*sin(3*y(x)/(2*x))*cos(y(x)/(2*x)) + 2*x*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*(x + 1)*sin(y(x)/(2*x))*cos(y(x)/(2*x))*cos(y(x)/x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out