Internal problem ID [9092]
Internal file name [OUTPUT/8027_Monday_June_06_2022_01_19_36_AM_74208493/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 758.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`x=_G(y,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (2 x +2+x^{3} y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{3} y^{2}-y^{\prime } \ln \left (y\right ) x -2 x^{2} y^{\prime }+2 x y-y^{\prime } \ln \left (y\right )-y^{\prime } x +2 y+y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x^{3} y^{2}-2 x y-2 y}{-\ln \left (y\right ) x -2 x^{2}-\ln \left (y\right )-x +1} \end {array} \]
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 differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[0, y^2/(ln(y)+2*x-1)]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
dsolve(diff(y(x),x) = (2*x+2+x^3*y(x))/(ln(y(x))+2*x-1)*y(x)/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {6 \operatorname {LambertW}\left (-\frac {\left (-2 x^{3}+3 x^{2}+6 \ln \left (x +1\right )+6 c_{1} -6 x \right ) {\mathrm e}^{-2 x}}{6}\right )}{-2 x^{3}+3 x^{2}+6 \ln \left (x +1\right )+6 c_{1} -6 x} \]
✓ Solution by Mathematica
Time used: 60.52 (sec). Leaf size: 459
DSolve[y'[x] == (y[x]*(2 + 2*x + x^3*y[x]))/((1 + x)*(-1 + 2*x + Log[y[x]])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {6 W\left (-\frac {1}{6} \sqrt [6]{e^{-12 x} \left (2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1\right ){}^6}\right )}{2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1} \\ y(x)\to \frac {6 W\left (\frac {1}{6} \sqrt [6]{e^{-12 x} \left (2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1\right ){}^6}\right )}{2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1} \\ y(x)\to \frac {6 W\left (-\frac {1}{6} \sqrt [3]{-1} \sqrt [6]{e^{-12 x} \left (2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1\right ){}^6}\right )}{2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1} \\ y(x)\to \frac {6 W\left (\frac {1}{6} \sqrt [3]{-1} \sqrt [6]{e^{-12 x} \left (2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1\right ){}^6}\right )}{2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1} \\ y(x)\to \frac {6 W\left (-\frac {1}{6} (-1)^{2/3} \sqrt [6]{e^{-12 x} \left (2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1\right ){}^6}\right )}{2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1} \\ y(x)\to \frac {6 W\left (\frac {1}{6} (-1)^{2/3} \sqrt [6]{e^{-12 x} \left (2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1\right ){}^6}\right )}{2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1} \\ \end{align*}