Internal problem ID [8176]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 596.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {-2 x^{2}+x +F \left (y+x^{2}-x \right )}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 39
dsolve(diff(y(x),x) = (-2*x^2+x+F(y(x)+x^2-x))/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = -x^{2}+x +\RootOf \left (F \left (\textit {\_Z} \right )\right ) \\ y \relax (x ) = -x^{2}+\RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1}\right )+x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.217 (sec). Leaf size: 156
DSolve[y'[x] == (x - 2*x^2 + F[-x + x^2 + y[x]])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (x^2-x+K[2]\right ) \int _1^x\left (\frac {2 K[1] F'\left (K[1]^2-K[1]+K[2]\right )}{F\left (K[1]^2-K[1]+K[2]\right )^2}-\frac {F'\left (K[1]^2-K[1]+K[2]\right )}{F\left (K[1]^2-K[1]+K[2]\right )^2}\right )dK[1]+1}{F\left (x^2-x+K[2]\right )}dK[2]+\int _1^x\left (-\frac {2 K[1]}{F\left (K[1]^2-K[1]+y(x)\right )}+\frac {1}{F\left (K[1]^2-K[1]+y(x)\right )}+\frac {1}{K[1]}\right )dK[1]=c_1,y(x)\right ] \]