Internal problem ID [8482]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 902.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {-x y^{2}+x^{3}-x -y^{6}+3 y^{4} x^{2}-3 x^{4} y^{2}+x^{6}}{\left (-y^{2}+x^{2}-1\right ) y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 175
dsolve(diff(y(x),x) = (-x*y(x)^2+x^3-x-y(x)^6+3*x^2*y(x)^4-3*x^4*y(x)^2+x^6)/(-y(x)^2+x^2-1)/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {\left (c_{1}+3 x \right ) \left (4 x^{2} c_{1}+12 x^{3}+\sqrt {-12 c_{1}-36 x +9}-3\right )}}{2 \left (c_{1}+3 x \right )} \\ y \relax (x ) = \frac {\sqrt {\left (c_{1}+3 x \right ) \left (4 x^{2} c_{1}+12 x^{3}+\sqrt {-12 c_{1}-36 x +9}-3\right )}}{2 c_{1}+6 x} \\ y \relax (x ) = -\frac {\sqrt {-\left (c_{1}+3 x \right ) \left (-4 x^{2} c_{1}-12 x^{3}+\sqrt {-12 c_{1}-36 x +9}+3\right )}}{2 \left (c_{1}+3 x \right )} \\ y \relax (x ) = \frac {\sqrt {-\left (c_{1}+3 x \right ) \left (-4 x^{2} c_{1}-12 x^{3}+\sqrt {-12 c_{1}-36 x +9}+3\right )}}{2 c_{1}+6 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 7.801 (sec). Leaf size: 215
DSolve[y'[x] == (-x + x^3 + x^6 - x*y[x]^2 - 3*x^4*y[x]^2 + 3*x^2*y[x]^4 - y[x]^6)/(y[x]*(-1 + x^2 - y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} \sqrt {-\frac {4 x^2 (-x+c_1)+\sqrt {-4 x+1+4 c_1}+1}{x-c_1}} \\ y(x)\to \frac {1}{2} \sqrt {-\frac {4 x^2 (-x+c_1)+\sqrt {-4 x+1+4 c_1}+1}{x-c_1}} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {4 x^2 (x-c_1)+\sqrt {-4 x+1+4 c_1}-1}{x-c_1}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {4 x^2 (x-c_1)+\sqrt {-4 x+1+4 c_1}-1}{x-c_1}} \\ y(x)\to -\sqrt {x^2} \\ y(x)\to \sqrt {x^2} \\ \end{align*}