Internal problem ID [8179]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 599.
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 {-x +F \left (y^{2}+x^{2}\right )}{y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.468 (sec). Leaf size: 87
dsolve(diff(y(x),x) = (-x+F(y(x)^2+x^2))/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \sqrt {-x^{2}+\RootOf \left (F \left (\textit {\_Z} \right )\right )} \\ y \relax (x ) = -\sqrt {-x^{2}+\RootOf \left (F \left (\textit {\_Z} \right )\right )} \\ y \relax (x ) = \sqrt {-x^{2}+\RootOf \left (-2 x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +2 c_{1}\right )} \\ y \relax (x ) = -\sqrt {-x^{2}+\RootOf \left (-2 x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +2 c_{1}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.142 (sec). Leaf size: 95
DSolve[y'[x] == (-x + F[x^2 + y[x]^2])/y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{F\left (x^2+K[2]^2\right )}-\int _1^x\frac {2 K[1] K[2] F'\left (K[1]^2+K[2]^2\right )}{F\left (K[1]^2+K[2]^2\right )^2}dK[1]\right )dK[2]+\int _1^x\left (1-\frac {K[1]}{F\left (K[1]^2+y(x)^2\right )}\right )dK[1]=c_1,y(x)\right ] \]