Internal problem ID [8162]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 582.
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 {1+F \left (\frac {y a x +1}{a x}\right ) a \,x^{2}}{a \,x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 48
dsolve(diff(y(x),x) = (1+F((y(x)*a*x+1)/a/x)*a*x^2)/a/x^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\RootOf \left (F \left (\textit {\_Z} \right )\right ) a x -1}{a x} \\ y \relax (x ) = \frac {\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1}\right ) a x -1}{a x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.213 (sec). Leaf size: 142
DSolve[y'[x] == (1 + a*x^2*F[(1 + a*x*y[x])/(a*x)])/(a*x^2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {a x K[2]+1}{a x}\right ) \int _1^x\frac {F'\left (\frac {a K[1] K[2]+1}{a K[1]}\right )}{a F\left (\frac {a K[1] K[2]+1}{a K[1]}\right )^2 K[1]^2}dK[1]-1}{F\left (\frac {a x K[2]+1}{a x}\right )}dK[2]+\int _1^x\left (-1-\frac {1}{a K[1]^2 F\left (\frac {a K[1] y(x)+1}{a K[1]}\right )}\right )dK[1]=c_1,y(x)\right ] \]