Internal problem ID [8416]
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.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (x -y\right ) \left (1+y\right )}{x \left (x y+x -y\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 102
dsolve(diff(y(x),x) = y(x)*(x-y(x))*(y(x)+1)/x/(x*y(x)+x-y(x)),y(x), singsol=all)
\[ y \relax (x ) = -\frac {{\mathrm e}^{\RootOf \left (-\ln \left (\frac {{\mathrm e}^{\textit {\_Z}}}{2}+\frac {9}{2}\right ) {\mathrm e}^{\textit {\_Z}}+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} x +9\right )} x}{{\mathrm e}^{\RootOf \left (-\ln \left (\frac {{\mathrm e}^{\textit {\_Z}}}{2}+\frac {9}{2}\right ) {\mathrm e}^{\textit {\_Z}}+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} x +9\right )} x -{\mathrm e}^{\RootOf \left (-\ln \left (\frac {{\mathrm e}^{\textit {\_Z}}}{2}+\frac {9}{2}\right ) {\mathrm e}^{\textit {\_Z}}+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} x +9\right )}-9} \]
✓ Solution by Mathematica
Time used: 8.3 (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]
\[ \text {Solve}\left [\frac {1}{9} 2^{2/3} \left (\frac {\left (1-\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}\right ) \left (\frac {\left (\frac {x^6}{(x-1)^3}\right )^{2/3} (x-1)^2 ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}+2\right ) \left (\left (1-\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}\right ) \log \left (2^{2/3} \left (1-\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}\right )\right )+\left (\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}-1\right ) \log \left (2^{2/3} \left (\frac {\left (\frac {x^6}{(x-1)^3}\right )^{2/3} (x-1)^2 ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}+2\right )\right )-3\right )}{\frac {3 (x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} ((x+2) y(x)+x)}{x^4 ((x-1) y(x)+x)}-\frac {((x+2) y(x)+x)^3}{((x-1) y(x)+x)^3}-2}+\frac {\left (\frac {x^6}{(x-1)^3}\right )^{2/3} (x-1)^2}{x^3}\right )=c_1,y(x)\right ] \]