Internal problem ID [8549]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 969.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [NONE]
Solve \begin {gather*} \boxed {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 )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 27
dsolve(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)/(x+1),y(x), singsol=all)
\[ y \relax (x ) = \frac {\arccos \left (\frac {x^{2} c_{1}+x^{2}+2 x +1}{\left (x +1\right )^{2}}\right ) x}{2} \]
✓ Solution by Mathematica
Time used: 22.542 (sec). Leaf size: 24
DSolve[y'[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)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \text {ArcSin}\left (\frac {e^{c_1} x}{x+1}\right ) \\ y(x)\to 0 \\ \end{align*}