Internal problem ID [8424]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 844.
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 (y x +x +y\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 97
dsolve(diff(y(x),x) = y(x)*(x+y(x))*(y(x)+1)/x/(x*y(x)+x+y(x)),y(x), singsol=all)
\[ y \relax (x ) = -\frac {{\mathrm e}^{\RootOf \left (-\ln \left (\frac {{\mathrm e}^{\textit {\_Z}}}{2}+\frac {9}{2}\right ) {\mathrm e}^{\textit {\_Z}}+3 \,{\mathrm e}^{\textit {\_Z}} c_{1}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{\textit {\_Z}}+9\right )} x}{{\mathrm e}^{\RootOf \left (-\ln \left (\frac {{\mathrm e}^{\textit {\_Z}}}{2}+\frac {9}{2}\right ) {\mathrm e}^{\textit {\_Z}}+3 \,{\mathrm e}^{\textit {\_Z}} c_{1}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{\textit {\_Z}}+9\right )} x +{\mathrm e}^{\RootOf \left (-\ln \left (\frac {{\mathrm e}^{\textit {\_Z}}}{2}+\frac {9}{2}\right ) {\mathrm e}^{\textit {\_Z}}+3 \,{\mathrm e}^{\textit {\_Z}} c_{1}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{\textit {\_Z}}+9\right )}+9} \]
✓ Solution by Mathematica
Time used: 8.142 (sec). Leaf size: 386
DSolve[y'[x] == (y[x]*(1 + y[x])*(x + y[x]))/(x*(x + y[x] + x*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2^{2/3} \left (1-\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}\right ) \left (\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}+2\right ) \left (\left (1-\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}\right ) \log \left (2^{2/3} \left (1-\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}\right )\right )+\left (\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}-1\right ) \log \left (2^{2/3} \left (\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}+2\right )\right )-3\right )}{9 \left (\frac {3 \left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}-\frac {((x-2) y(x)+x)^3}{((x+1) y(x)+x)^3}-2\right )}=\frac {2^{2/3} \left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2}{9 x^3}+c_1,y(x)\right ] \]