Internal problem ID [8353]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 773.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class D], _rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x y+x +y^{2}}{\left (x -1\right ) \left (x +y\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.85 (sec). Leaf size: 48
dsolve(diff(y(x),x) = 1/(x-1)*(x*y(x)+x+y(x)^2)/(x+y(x)),y(x), singsol=all)
\[ y \relax (x ) = \frac {\sqrt {3}\, x \tan \left (\RootOf \left (-\sqrt {3}\, \ln \left (\frac {3 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}{4 \left (x -1\right )^{2}}\right )+2 \sqrt {3}\, c_{1}-2 \textit {\_Z} \right )\right )}{2}-\frac {x}{2} \]
✓ Solution by Mathematica
Time used: 0.151 (sec). Leaf size: 61
DSolve[y'[x] == (x + x*y[x] + y[x]^2)/((-1 + x)*(x + y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\text {ArcTan}\left (\frac {\frac {2 y(x)}{x}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+\frac {y(x)}{x}+1\right )=\log (1-x)-\log (x)+c_1,y(x)\right ] \]