Internal problem ID [9108]
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`]]
\[ \boxed {y^{\prime }-\frac {y x +x +y^{2}}{\left (x -1\right ) \left (x +y\right )}=0} \]
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 58
dsolve(diff(y(x),x) = 1/(x-1)*(x*y(x)+x+y(x)^2)/(x+y(x)),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x \left (\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (2 \sqrt {3}\, \ln \left (2\right )-\sqrt {3}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} x^{2}}{\left (x -1\right )^{2}}\right )-\sqrt {3}\, \ln \left (3\right )+2 \sqrt {3}\, c_{1} -2 \textit {\_Z} \right )\right )-1\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.146 (sec). Leaf size: 61
DSolve[y'[x] == (x + x*y[x] + y[x]^2)/((-1 + x)*(x + y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\arctan \left (\frac {\frac {2 y(x)}{x}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+\frac {y(x)}{x}+1\right )=\log (1-x)-\log (x)+c_1,y(x)\right ] \]