Internal problem ID [8483]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 903.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class D]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\sin \left (\frac {y}{x}\right ) \left (y+2 x^{2} \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )\right )}{2 \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 15
dsolve(diff(y(x),x) = 1/2*sin(y(x)/x)*(y(x)+2*x^2*sin(1/2*y(x)/x)*cos(1/2*y(x)/x))/sin(1/2*y(x)/x)/x/cos(1/2*y(x)/x),y(x), singsol=all)
\[ y \relax (x ) = \left (\pi -\arccos \left (\tanh \left (c_{1}+x \right )\right )\right ) x \]
✓ Solution by Mathematica
Time used: 4.257 (sec). Leaf size: 41
DSolve[y'[x] == (Csc[y[x]/(2*x)]*Sec[y[x]/(2*x)]*Sin[y[x]/x]*(2*x^2*Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)] + y[x]))/(2*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2 x \text {ArcTan}\left (e^{x+c_1}\right ) \\ y(x)\to 0 \\ y(x)\to \pi x (-1)^{\left \lfloor \frac {1}{2}-\frac {\Im (x)}{\pi }\right \rfloor } \\ \end{align*}