Internal problem ID [8361]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 781.
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}+x^{3}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
dsolve(diff(y(x),x) = 1/(6*y(x)^2+x)*(x^4+x^3+x+3*y(x)^2)*y(x)/x,y(x), singsol=all)
\[ \frac {1}{\frac {1}{y \relax (x )^{2}}+\frac {6}{x}} = \frac {\left ({\mathrm e}^{\RootOf \left (2 x^{3} {\mathrm e}^{\textit {\_Z}}+3 x^{2} {\mathrm e}^{\textit {\_Z}}-3 \,{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{\textit {\_Z}}+9}{2 x}\right )+9 \,{\mathrm e}^{\textit {\_Z}} c_{1}+3 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+27\right )}+9\right ) x}{54} \]
✓ Solution by Mathematica
Time used: 52.567 (sec). Leaf size: 87
DSolve[y'[x] == (y[x]*(x + x^3 + x^4 + 3*y[x]^2))/(x*(x + 6*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {x} \sqrt {\text {ProductLog}\left (6 x e^{\frac {2 x^3}{3}+x^2+2 c_1}\right )}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {x} \sqrt {\text {ProductLog}\left (6 x e^{\frac {2 x^3}{3}+x^2+2 c_1}\right )}}{\sqrt {6}} \\ y(x)\to 0 \\ \end{align*}