Internal problem ID [9239]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 904.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class D`]]
\[ \boxed {y^{\prime }-\frac {\sin \left (\frac {y}{x}\right ) \left (y+2 \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x^{3}\right )}{2 \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right )}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 65
dsolve(diff(y(x),x) = 1/2*sin(y(x)/x)*(y(x)+2*x^3*cos(1/2*y(x)/x)*sin(1/2*y(x)/x))/sin(1/2*y(x)/x)/x/cos(1/2*y(x)/x),y(x), singsol=all)
\[ y \left (x \right ) = \arctan \left (\frac {2 \,{\mathrm e}^{-\frac {x^{2}}{2}}}{c_{1} \left (\frac {{\mathrm e}^{-x^{2}}}{c_{1}^{2}}+1\right )}, \frac {\frac {{\mathrm e}^{-x^{2}}}{c_{1}^{2}}-1}{\frac {{\mathrm e}^{-x^{2}}}{c_{1}^{2}}+1}\right ) x \]
✓ Solution by Mathematica
Time used: 0.408 (sec). Leaf size: 62
DSolve[y'[x] == (Csc[y[x]/(2*x)]*Sec[y[x]/(2*x)]*Sin[y[x]/x]*(2*x^3*Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)] + y[x]))/(2*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \arccos \left (-\tanh \left (\frac {x^2}{2}+c_1\right )\right ) y(x)\to x \arccos \left (-\tanh \left (\frac {x^2}{2}+c_1\right )\right ) y(x)\to 0 y(x)\to -\pi x y(x)\to \pi x \end{align*}