Internal problem ID [8544]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 964.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {8 x \left (a -1\right ) \left (a +1\right )}{8+2 x^{4}+2 y^{4}-8 y+4 x^{2} y^{2}-4 x^{6} a^{2}-8 a^{2}+x^{6}+y^{6}-2 y^{4} a^{2}-8 y^{2} a^{2} x^{2}+6 a^{4} x^{4}+3 x^{2} y^{4}-6 y^{4} a^{2} x^{2}-9 y^{2} a^{2} x^{4}-y^{6} a^{2}-2 a^{6} x^{4}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}+4 a^{4} y^{2} x^{2}+9 y^{2} a^{4} x^{4}-3 a^{6} y^{2} x^{4}+3 a^{4} y^{4} x^{2}+3 x^{4} y^{2}-6 a^{2} x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.718 (sec). Leaf size: 652
dsolve(diff(y(x),x) = -8*x*(a-1)*(a+1)/(8-a^2*y(x)^6+a^8*x^6-4*a^6*x^6+6*a^4*x^6-2*a^2*y(x)^4-2*a^6*x^4+6*a^4*x^4-6*a^2*x^4-4*a^2*x^6+y(x)^6+x^6-8*a^2+3*a^4*y(x)^4*x^2-3*a^6*y(x)^2*x^4+9*y(x)^2*a^4*x^4-9*y(x)^2*a^2*x^4+4*a^4*y(x)^2*x^2-6*y(x)^4*a^2*x^2-8*y(x)^2*a^2*x^2+3*x^2*y(x)^4+3*x^4*y(x)^2+4*x^2*y(x)^2+2*x^4+2*y(x)^4-8*y(x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {-3 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-3 a^{2} x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+3 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (116+12 \sqrt {93}\right )^{\frac {2}{3}}+2 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+4\right )}}{3 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {-3 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-3 a^{2} x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+3 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (116+12 \sqrt {93}\right )^{\frac {2}{3}}+2 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+4\right )}}{3 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\sqrt {-6 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-6 a^{2} x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+i \left (116+12 \sqrt {93}\right )^{\frac {2}{3}} \sqrt {3}+6 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}-\left (116+12 \sqrt {93}\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}-4\right )}}{6 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {-6 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-6 a^{2} x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+i \left (116+12 \sqrt {93}\right )^{\frac {2}{3}} \sqrt {3}+6 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}-\left (116+12 \sqrt {93}\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}-4\right )}}{6 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\sqrt {6}\, \sqrt {\left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (6 a^{2} x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+i \left (116+12 \sqrt {93}\right )^{\frac {2}{3}} \sqrt {3}-6 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (116+12 \sqrt {93}\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+4\right )}}{6 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {6}\, \sqrt {\left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (6 a^{2} x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+i \left (116+12 \sqrt {93}\right )^{\frac {2}{3}} \sqrt {3}-6 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (116+12 \sqrt {93}\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}+4\right )}}{6 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ \frac {y \relax (x )}{\left (a +1\right ) \left (a -1\right )}+\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+2 \textit {\_Z}^{2}+8\right )}{\sum }\frac {\ln \left (-a^{2} x^{2}+x^{2}+y \relax (x )^{2}-\textit {\_R} \right )}{3 \textit {\_R}^{2}+4 \textit {\_R}}\right )}{a^{4}-2 a^{2}+1}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.366 (sec). Leaf size: 264
DSolve[y'[x] == (-8*(-1 + a)*(1 + a)*x)/(8 - 8*a^2 + 2*x^4 - 6*a^2*x^4 + 6*a^4*x^4 - 2*a^6*x^4 + x^6 - 4*a^2*x^6 + 6*a^4*x^6 - 4*a^6*x^6 + a^8*x^6 - 8*y[x] + 4*x^2*y[x]^2 - 8*a^2*x^2*y[x]^2 + 4*a^4*x^2*y[x]^2 + 3*x^4*y[x]^2 - 9*a^2*x^4*y[x]^2 + 9*a^4*x^4*y[x]^2 - 3*a^6*x^4*y[x]^2 + 2*y[x]^4 - 2*a^2*y[x]^4 + 3*x^2*y[x]^4 - 6*a^2*x^2*y[x]^4 + 3*a^4*x^2*y[x]^4 + y[x]^6 - a^2*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {y(x)}{(a-1) (a+1)}-\frac {8 \text {RootSum}\left [-\text {$\#$1}^3 a^6+3 \text {$\#$1}^3 a^4-3 \text {$\#$1}^3 a^2+\text {$\#$1}^3+3 \text {$\#$1}^2 a^4 y(x)^2+2 \text {$\#$1}^2 a^4-6 \text {$\#$1}^2 a^2 y(x)^2-4 \text {$\#$1}^2 a^2+3 \text {$\#$1}^2 y(x)^2+2 \text {$\#$1}^2-3 \text {$\#$1} a^2 y(x)^4-4 \text {$\#$1} a^2 y(x)^2+3 \text {$\#$1} y(x)^4+4 \text {$\#$1} y(x)^2+y(x)^6+2 y(x)^4+8\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2 a^4-6 \text {$\#$1}^2 a^2+3 \text {$\#$1}^2-6 \text {$\#$1} a^2 y(x)^2-4 \text {$\#$1} a^2+6 \text {$\#$1} y(x)^2+4 \text {$\#$1}+3 y(x)^4+4 y(x)^2}\&\right ]}{(a-1) (a+1) \left (2-2 a^2\right )}=c_1,y(x)\right ] \]