Internal problem ID [8470]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 890.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x}{-y+1+y^{4}+2 x^{2} y^{2}+x^{4}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.118 (sec). Leaf size: 504
dsolve(diff(y(x),x) = x/(-y(x)+1+y(x)^4+2*x^2*y(x)^2+x^4+y(x)^6+3*x^2*y(x)^4+3*x^4*y(x)^2+x^6),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {-6 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (6 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 )}}{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 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 )}}{6 \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 (i \sqrt {3}\, \left (116+12 \sqrt {93}\right )^{\frac {2}{3}}+12 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}-4 i \sqrt {3}-\left (116+12 \sqrt {93}\right )^{\frac {2}{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 {-3 \left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (116+12 \sqrt {93}\right )^{\frac {2}{3}}+12 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}-4 i \sqrt {3}-\left (116+12 \sqrt {93}\right )^{\frac {2}{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 {3}\, \sqrt {\left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (116+12 \sqrt {93}\right )^{\frac {2}{3}}-12 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}-4 i \sqrt {3}+\left (116+12 \sqrt {93}\right )^{\frac {2}{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 {3}\, \sqrt {\left (116+12 \sqrt {93}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (116+12 \sqrt {93}\right )^{\frac {2}{3}}-12 x^{2} \left (116+12 \sqrt {93}\right )^{\frac {1}{3}}-4 i \sqrt {3}+\left (116+12 \sqrt {93}\right )^{\frac {2}{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 {\left (\int _{}^{x^{2}+y \relax (x )^{2}}\frac {1}{\textit {\_a}^{3}+\textit {\_a}^{2}+1}d \textit {\_a} \right )}{2}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.141 (sec). Leaf size: 103
DSolve[y'[x] == x/(1 + x^4 + x^6 - y[x] + 2*x^2*y[x]^2 + 3*x^4*y[x]^2 + y[x]^4 + 3*x^2*y[x]^4 + y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [y(x)-\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^3+3 \text {$\#$1}^2 y(x)^2+\text {$\#$1}^2+3 \text {$\#$1} y(x)^4+2 \text {$\#$1} y(x)^2+y(x)^6+y(x)^4+1\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2+6 \text {$\#$1} y(x)^2+2 \text {$\#$1}+3 y(x)^4+2 y(x)^2}\&\right ]=c_1,y(x)\right ] \]