Internal problem ID [8194]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 614.
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 \left (a -1\right ) \left (a +1\right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 107
dsolve(diff(y(x),x) = x*(a-1)*(a+1)/(y(x)+F(1/2*y(x)^2-1/2*a^2*x^2+1/2*x^2)*a^2-F(1/2*y(x)^2-1/2*a^2*x^2+1/2*x^2)),y(x), singsol=all)
\begin{align*} y \relax (x ) = \sqrt {x^{2} a^{2}-x^{2}+2 \RootOf \left (F \left (\textit {\_Z} \right )\right )} \\ y \relax (x ) = -\sqrt {x^{2} a^{2}-x^{2}+2 \RootOf \left (F \left (\textit {\_Z} \right )\right )} \\ \frac {y \relax (x )}{\left (a -1\right ) \left (a +1\right )}+\frac {\int _{}^{-x^{2} a^{2}+x^{2}+y \relax (x )^{2}}\frac {1}{F \left (\frac {\textit {\_a}}{2}\right )}d \textit {\_a}}{2 a^{4}-4 a^{2}+2}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.285 (sec). Leaf size: 177
DSolve[y'[x] == ((-1 + a)*(1 + a)*x)/(-F[x^2/2 - (a^2*x^2)/2 + y[x]^2/2] + a^2*F[x^2/2 - (a^2*x^2)/2 + y[x]^2/2] + y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{(a-1) (a+1) F\left (-\frac {1}{2} a^2 x^2+\frac {x^2}{2}+\frac {K[2]^2}{2}\right )}-\int _1^x\frac {K[1] K[2] F'\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )^2}dK[1]+1\right )dK[2]+\int _1^x-\frac {K[1]}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {y(x)^2}{2}\right )}dK[1]=c_1,y(x)\right ] \]