Internal problem ID [8175]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 595.
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 {F \left (\frac {x y^{2}+1}{x}\right )}{y x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.609 (sec). Leaf size: 117
dsolve(diff(y(x),x) = F((x*y(x)^2+1)/x)/y(x)/x^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {x \left (\RootOf \left (2 F \left (\textit {\_Z} \right )-1\right ) x -1\right )}}{x} \\ y \relax (x ) = -\frac {\sqrt {x \left (\RootOf \left (2 F \left (\textit {\_Z} \right )-1\right ) x -1\right )}}{x} \\ y \relax (x ) = \frac {\sqrt {x \left (\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{-1+2 F \left (\textit {\_a} \right )}d \textit {\_a} \right ) x +x c_{1}+1\right ) x -1\right )}}{x} \\ y \relax (x ) = -\frac {\sqrt {x \left (\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{-1+2 F \left (\textit {\_a} \right )}d \textit {\_a} \right ) x +x c_{1}+1\right ) x -1\right )}}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.244 (sec). Leaf size: 204
DSolve[y'[x] == F[(1 + x*y[x]^2)/x]/(x^2*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{2 F\left (\frac {x K[2]^2+1}{x}\right )-1}-\int _1^x\left (\frac {4 F\left (\frac {K[1] K[2]^2+1}{K[1]}\right ) K[2] F'\left (\frac {K[1] K[2]^2+1}{K[1]}\right )}{\left (2 F\left (\frac {K[1] K[2]^2+1}{K[1]}\right )-1\right )^2 K[1]^2}-\frac {2 K[2] F'\left (\frac {K[1] K[2]^2+1}{K[1]}\right )}{\left (2 F\left (\frac {K[1] K[2]^2+1}{K[1]}\right )-1\right ) K[1]^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {F\left (\frac {K[1] y(x)^2+1}{K[1]}\right )}{\left (2 F\left (\frac {K[1] y(x)^2+1}{K[1]}\right )-1\right ) K[1]^2}dK[1]=c_1,y(x)\right ] \]