Internal problem ID [8540]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 960.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {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}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 11
dsolve(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,y(x), singsol=all)
\[ y \relax (x ) = \arcsin \left ({\mathrm e}^{x} c_{1}\right ) x \]
✓ Solution by Mathematica
Time used: 8.156 (sec). Leaf size: 19
DSolve[y'[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,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \text {ArcSin}\left (e^{x+c_1}\right ) \\ y(x)\to 0 \\ \end{align*}