Internal problem ID [8950]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 615.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\[ \boxed {y^{\prime }-\frac {y}{x \left (-1+F \left (y x \right ) y\right )}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 38
dsolve(diff(y(x),x) = y(x)/x/(-1+F(x*y(x))*y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (F \left (x \textit {\_Z} \right )\right ) \\ -y \left (x \right )+\int _{}^{x y \left (x \right )}\frac {1}{F \left (\textit {\_a} \right ) \textit {\_a}}d \textit {\_a} -c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.261 (sec). Leaf size: 77
DSolve[y'[x] == y[x]/(x*(-1 + F[x*y[x]]*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\int _1^x\frac {F'(K[1] K[2])}{F(K[1] K[2])^2}dK[1]-\frac {1}{F(x K[2]) K[2]}+1\right )dK[2]+\int _1^x-\frac {1}{F(K[1] y(x)) K[1]}dK[1]=c_1,y(x)\right ] \]