Internal problem ID [8191]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 611.
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 x -y+F \left (x \left (x +y\right )\right )}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 44
dsolve(diff(y(x),x) = (-2*x-y(x)+F((x+y(x))*x))/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {-x^{2}+\RootOf \left (F \left (\textit {\_Z} \right )\right )}{x} \\ y \relax (x ) = \frac {-x^{2}+\RootOf \left (-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: 0.19 (sec). Leaf size: 191
DSolve[y'[x] == (-2*x + F[x*(x + y[x])] - y[x])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x+F(x (x+K[2])) \int _1^x\left (\frac {2 F'(K[1] (K[1]+K[2])) K[1]^2}{F(K[1] (K[1]+K[2]))^2}+\frac {(K[2]-F(K[1] (K[1]+K[2]))) F'(K[1] (K[1]+K[2])) K[1]}{F(K[1] (K[1]+K[2]))^2}-\frac {1-K[1] F'(K[1] (K[1]+K[2]))}{F(K[1] (K[1]+K[2]))}\right )dK[1]}{F(x (x+K[2]))}dK[2]+\int _1^x\left (-\frac {2 K[1]}{F(K[1] (K[1]+y(x)))}-\frac {y(x)-F(K[1] (K[1]+y(x)))}{F(K[1] (K[1]+y(x)))}\right )dK[1]=c_1,y(x)\right ] \]