Internal problem ID [8177]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 597.
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 {2 a}{x^{2} \left (-y+2 F \left (\frac {x y^{2}-4 a}{x}\right ) a \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.069 (sec). Leaf size: 78
dsolve(diff(y(x),x) = 2*a/x^2/(-y(x)+2*F((x*y(x)^2-4*a)/x)*a),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {x \left (\RootOf \left (F \left (\textit {\_Z} \right )\right ) x +4 a \right )}}{x} \\ y \relax (x ) = -\frac {\sqrt {x \left (\RootOf \left (F \left (\textit {\_Z} \right )\right ) x +4 a \right )}}{x} \\ -\frac {y \relax (x )}{2 a}+\frac {\int _{}^{y \relax (x )^{2}-\frac {4 a}{x}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a}}{8 a^{2}}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.308 (sec). Leaf size: 130
DSolve[y'[x] == (2*a)/(x^2*(2*a*F[(-4*a + x*y[x]^2)/x] - y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{2 a F\left (\frac {x K[2]^2-4 a}{x}\right )}-\int _1^x\frac {2 K[2] F'\left (\frac {K[1] K[2]^2-4 a}{K[1]}\right )}{F\left (\frac {K[1] K[2]^2-4 a}{K[1]}\right )^2 K[1]^2}dK[1]+1\right )dK[2]+\int _1^x-\frac {1}{F\left (\frac {K[1] y(x)^2-4 a}{K[1]}\right ) K[1]^2}dK[1]=c_1,y(x)\right ] \]