Internal problem ID [8414]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 834.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (x^{4}+3 x y^{2}+3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 60
dsolve(diff(y(x),x) = (x^4+3*x*y(x)^2+3*y(x)^2)/(6*y(x)^2+x)*y(x)/x/(x+1),y(x), singsol=all)
\[ \frac {1}{\frac {1}{y \relax (x )^{2}}+\frac {6}{x}} = \frac {\left ({\mathrm e}^{\RootOf \left (x^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {x \left ({\mathrm e}^{\textit {\_Z}}+9\right )}{2 \left (x +1\right )^{2}}\right )+3 \,{\mathrm e}^{\textit {\_Z}} c_{1}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-2 x \,{\mathrm e}^{\textit {\_Z}}+9\right )}+9\right ) x}{54} \]
✓ Solution by Mathematica
Time used: 46.154 (sec). Leaf size: 93
DSolve[y'[x] == (y[x]*(x^4 + 3*y[x]^2 + 3*x*y[x]^2))/(x*(1 + x)*(x + 6*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {x} \sqrt {\text {ProductLog}\left (\frac {6 (x+1)^2 e^{(x-2) x-3+2 c_1}}{x}\right )}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {x} \sqrt {\text {ProductLog}\left (\frac {6 (x+1)^2 e^{(x-2) x-3+2 c_1}}{x}\right )}}{\sqrt {6}} \\ y(x)\to 0 \\ \end{align*}