Internal problem ID [8492]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 912.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 a x}{-x^{3} y+2 x^{3} a +2 a y^{4} x^{3}-16 y^{2} a^{2} x^{2}+32 a^{3} x +2 a y^{6} x^{3}-24 y^{4} a^{2} x^{2}+96 y^{2} x \,a^{3}-128 a^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 1997.156 (sec). Leaf size: 809
dsolve(diff(y(x),x) = 2*a*x/(-x^3*y(x)+2*x^3*a+2*a*y(x)^4*x^3-16*y(x)^2*a^2*x^2+32*a^3*x+2*a*y(x)^6*x^3-24*y(x)^4*a^2*x^2+96*y(x)^2*x*a^3-128*a^4),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {6}\, \sqrt {x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}} \left (\left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}+24 a \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}-2 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}+4 x^{2}\right )}}{6 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {6}\, \sqrt {x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}} \left (\left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}+24 a \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}-2 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}+4 x^{2}\right )}}{6 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\sqrt {-3 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}+\left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}-48 a \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}+4 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}+4 x^{2}\right )}}{6 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {-3 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}+\left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}-48 a \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}+4 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}+4 x^{2}\right )}}{6 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\sqrt {3}\, \sqrt {x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}-\left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}+48 a \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}-4 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}-4 x^{2}\right )}}{6 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {3}\, \sqrt {x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}-\left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {2}{3}}+48 a \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}-4 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}-4 x^{2}\right )}}{6 x \left (-116 x^{3}+12 \sqrt {93}\, x^{3}\right )^{\frac {1}{3}}} \\ -\frac {y \relax (x )}{2 a}+\frac {\int _{}^{y \relax (x )^{2}-\frac {4 a}{x}}\frac {1}{\textit {\_a}^{3}+\textit {\_a}^{2}+1}d \textit {\_a}}{8 a^{2}}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.865 (sec). Leaf size: 201
DSolve[y'[x] == (2*a*x)/(-128*a^4 + 32*a^3*x + 2*a*x^3 - x^3*y[x] + 96*a^3*x*y[x]^2 - 16*a^2*x^2*y[x]^2 - 24*a^2*x^2*y[x]^4 + 2*a*x^3*y[x]^4 + 2*a*x^3*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\text {RootSum}\left [-\text {$\#$1}^3 y(x)^6-\text {$\#$1}^3 y(x)^4-\text {$\#$1}^3+12 \text {$\#$1}^2 a y(x)^4+8 \text {$\#$1}^2 a y(x)^2-48 \text {$\#$1} a^2 y(x)^2-16 \text {$\#$1} a^2+64 a^3\&,\frac {\text {$\#$1} \log (x-\text {$\#$1})}{3 \text {$\#$1}^2 y(x)^6+3 \text {$\#$1}^2 y(x)^4+3 \text {$\#$1}^2-24 \text {$\#$1} a y(x)^4-16 \text {$\#$1} a y(x)^2+48 a^2 y(x)^2+16 a^2}\&\right ]-\frac {\text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+1\&,\frac {\log \left (y(x)^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2+2 \text {$\#$1}}\&\right ]}{4 a}+y(x)=c_1,y(x)\right ] \]