Internal problem ID [8196]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 616.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {-x^{2}+2 y x^{3}-F \left (\left (y x -1\right ) x \right )}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 38
dsolve(diff(y(x),x) = -1/x^4*(-x^2+2*x^3*y(x)-F((x*y(x)-1)*x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x +\RootOf \left (F \left (\textit {\_Z} \right )\right )}{x^{2}} \\ y \relax (x ) = \frac {\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} \right ) x +x c_{1}+1\right )+x}{x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.327 (sec). Leaf size: 177
DSolve[y'[x] == (x^2 + F[x*(-1 + x*y[x])] - 2*x^3*y[x])/x^4,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x^2+F(x (x K[2]-1)) \int _1^x\left (\frac {2 K[2] F'(K[1] (K[1] K[2]-1)) K[1]^3}{F(K[1] (K[1] K[2]-1))^2}-\frac {F'(K[1] (K[1] K[2]-1)) K[1]^2}{F(K[1] (K[1] K[2]-1))^2}-\frac {2 K[1]}{F(K[1] (K[1] K[2]-1))}\right )dK[1]}{F(x (x K[2]-1))}dK[2]+\int _1^x\left (-\frac {2 K[1] y(x)}{F(K[1] (K[1] y(x)-1))}+\frac {1}{F(K[1] (K[1] y(x)-1))}+\frac {1}{K[1]^2}\right )dK[1]=c_1,y(x)\right ] \]