Internal problem ID [9168]
Internal file name [OUTPUT/8103_Monday_June_06_2022_01_46_06_AM_98301138/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (x^{4}+3 y^{2} x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (1+x \right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 6 y^{\prime } y^{2} x^{2}-x^{4} y+6 y^{\prime } y^{2} x +y^{\prime } x^{3}-3 x y^{3}+x^{2} y^{\prime }-3 y^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{4} y+3 x y^{3}+3 y^{3}}{6 y^{2} x^{2}+6 y^{2} x +x^{3}+x^{2}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE equivalence obtained to this Abel ODE: diff(y(x),x) = 2*x^2/(x+1)*y(x)-12*(2*x^3-x-1)/x^2/(x+1)*y(x)^2+36*(2*x^3-x-1)/(x+1)/x^3*y(x) trying to solve the Abel ODE ... <- Abel successful equivalence to an Abel ODE successful, Abel ODE has been solved`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 68
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)
\[ \frac {y \left (x \right )^{2} x}{6 y \left (x \right )^{2}+x} = \frac {\left ({\mathrm e}^{\operatorname {RootOf}\left (x^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {x \left ({\mathrm e}^{\textit {\_Z}}+9\right )}{\left (x +1\right )^{2}}\right )+{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -2 x \,{\mathrm e}^{\textit {\_Z}}+9\right )}+9\right ) x}{54} \]
✓ Solution by Mathematica
Time used: 11.9 (sec). Leaf size: 95
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 {W\left (\frac {6 (x+1)^2 e^{x^2-2 x-3+2 c_1}}{x}\right )}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {x} \sqrt {W\left (\frac {6 (x+1)^2 e^{x^2-2 x-3+2 c_1}}{x}\right )}}{\sqrt {6}} \\ y(x)\to 0 \\ \end{align*}