Internal problem ID [8168]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 588.
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 (-\left (x -y\right ) \left (x +y\right )\right )}{y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.485 (sec). Leaf size: 79
dsolve(diff(y(x),x) = (x+F(-(x-y(x))*(x+y(x))))/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: 0.163 (sec). Leaf size: 109
DSolve[y'[x] == (x + F[(-x + y[x])*(x + y[x])])/y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{F((K[2]-x) (x+K[2]))}-\int _1^x-\frac {2 K[1] K[2] F'((K[2]-K[1]) (K[1]+K[2]))}{F((K[2]-K[1]) (K[1]+K[2]))^2}dK[1]\right )dK[2]+\int _1^x\left (\frac {K[1]}{F((y(x)-K[1]) (K[1]+y(x)))}+1\right )dK[1]=c_1,y(x)\right ] \]