Internal problem ID [8373]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 793.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {y \left (x y+1\right )}{x \left (x y+1-y\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 32
dsolve(diff(y(x),x) = -1/x*y(x)*(x*y(x)+1)/(x*y(x)+1-y(x)),y(x), singsol=all)
\[ y \relax (x ) = -\frac {2 \,{\mathrm e}^{-\LambertW \left (-\frac {2 \left (x -1\right ) {\mathrm e}^{3 c_{1}} {\mathrm e}^{-1}}{x}\right )+3 c_{1}-1}}{x} \]
✓ Solution by Mathematica
Time used: 8.872 (sec). Leaf size: 399
DSolve[y'[x] == -((y[x]*(1 + x*y[x]))/(x*(1 - y[x] + x*y[x]))),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\sqrt [3]{-2} \left (\frac {2^{2/3} ((x-1) y(x)-2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+(-2)^{2/3}\right ) \left (\frac {-x y(x)+y(x)+2}{\sqrt [3]{2} \sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+(-2)^{2/3}\right ) \left (\left (\frac {\sqrt [3]{-1} (-x y(x)+y(x)+2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+1\right ) \left (-\log \left (\frac {2^{2/3} ((x-1) y(x)-2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+(-2)^{2/3}\right )\right )+\left (\frac {\sqrt [3]{-1} (-x y(x)+y(x)+2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+1\right ) \log \left (\frac {2^{2/3} (-x y(x)+y(x)+2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+2 (-2)^{2/3}\right )+3\right )}{9 \left (\frac {((x-1) y(x)-2)^3}{((x-1) y(x)+1)^3}+\frac {3 \sqrt [3]{-1} ((x-1) y(x)-2)}{\left (-\frac {1}{(x-1)^3}\right )^{4/3} (x-1)^4 ((x-1) y(x)+1)}+2\right )}=\frac {1}{9} 2^{2/3} \left (-\frac {1}{(x-1)^3}\right )^{2/3} (x-1)^2 (\log (1-x)-\log (x))+c_1,y(x)\right ] \]