Internal problem ID [9061]
Internal file name [OUTPUT/7996_Monday_June_06_2022_01_09_39_AM_4705602/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 727.
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.
\[ \boxed {y^{\prime }-\frac {\left (2 x +2+y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{\prime } \ln \left (y\right ) x -2 y^{\prime } x^{2}+y^{2}+2 x y-y^{\prime } \ln \left (y\right )-y^{\prime } x +2 y+y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{2}-2 x y-2 y}{-\ln \left (y\right ) x -2 x^{2}-\ln \left (y\right )-x +1} \end {array} \]
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`[(x+1)*y/(2*x+2+y), 0], [-1/(2*x+2+y)*y, 1/(ln(y)+2*x-1)/(x+1)*x*y^2]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 37
dsolve(diff(y(x),x) = (2*x+2+y(x))/(ln(y(x))+2*x-1)*y(x)/(x+1),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -2 x -2 \\ y \left (x \right ) &= \frac {\operatorname {LambertW}\left (\left (\ln \left (x +1\right )-c_{1} \right ) {\mathrm e}^{-2 x}\right )}{\ln \left (x +1\right )-c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.295 (sec). Leaf size: 29
DSolve[y'[x] == (y[x]*(2 + 2*x + y[x]))/((1 + x)*(-1 + 2*x + Log[y[x]])),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {W\left (e^{-2 x} (\log (x+1)+c_1)\right )}{\log (x+1)+c_1} \]