Internal problem ID [9170]
Internal file name [OUTPUT/8105_Monday_June_06_2022_01_46_34_AM_19026451/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 836.
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, [_Abel, `2nd type`, `class C`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {y \left (x -y\right ) \left (y+1\right )}{x \left (y x +x -y\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y x^{2}-y^{\prime } x y+y^{\prime } x^{2}+y^{3}-y^{2} x +y^{2}-y x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{3}+y^{2} x -y^{2}+y x}{y x^{2}-y x +x^{2}} \end {array} \]
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 trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 79
dsolve(diff(y(x),x) = y(x)*(x-y(x))*(y(x)+1)/x/(x*y(x)+x-y(x)),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {{\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-{\mathrm e}^{\textit {\_Z}} \ln \left ({\mathrm e}^{\textit {\_Z}}+9\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+9\right )} x}{-9+\left (x -1\right ) {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-{\mathrm e}^{\textit {\_Z}} \ln \left ({\mathrm e}^{\textit {\_Z}}+9\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+9\right )}} \]
✓ Solution by Mathematica
Time used: 9.315 (sec). Leaf size: 379
DSolve[y'[x] == ((x - y[x])*y[x]*(1 + y[x]))/(x*(x - y[x] + x*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{9} 2^{2/3} \left (\frac {\left (1-\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}\right ) \left (\frac {\left (\frac {x^6}{(x-1)^3}\right )^{2/3} (x-1)^2 ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}+2\right ) \left (\left (1-\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}\right ) \log \left (2^{2/3} \left (1-\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}\right )\right )+\left (\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}-1\right ) \log \left (2^{2/3} \left (\frac {\left (\frac {x^6}{(x-1)^3}\right )^{2/3} (x-1)^2 ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}+2\right )\right )-3\right )}{\frac {3 (x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}-\frac {((x+2) y(x)+x)^3}{((x-1) y(x)+x)^3}-2}+\frac {\left (\frac {x^6}{(x-1)^3}\right )^{2/3} (x-1)^2}{x^3}\right )=c_1,y(x)\right ] \]