Internal problem ID [9132]
Internal file name [OUTPUT/8067_Monday_June_06_2022_01_38_38_AM_32350857/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 798.
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]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (x +y+1\right ) y}{\left (2 y^{3}+y+x \right ) \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y^{\prime } y^{3} x +2 y^{\prime } y^{3}+y^{\prime } y x +y^{\prime } x^{2}+y^{\prime } y+y^{\prime } x -y^{2}-y x -y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2}+y x +y}{2 y^{3} x +2 y^{3}+y x +x^{2}+y+x} \end {array} \]
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 = 2`[0, (x*y^2+y^2)/(2*y^3+x+y)/(x+1)]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 30
dsolve(diff(y(x),x) = 1/(2*y(x)^3+y(x)+x)*(x+y(x)+1)*y(x)/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{3 \textit {\_Z}}+\ln \left (x +1\right ) {\mathrm e}^{\textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} \textit {\_Z} +x \right )} \]
✓ Solution by Mathematica
Time used: 0.323 (sec). Leaf size: 27
DSolve[y'[x] == (y[x]*(1 + x + y[x]))/((1 + x)*(x + y[x] + 2*y[x]^3)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [y(x)^2-\frac {x}{y(x)}+\log (y(x))-\log (x+1)=c_1,y(x)\right ] \]