Internal problem ID [8340]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 760.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (x y^{2}+1\right )^{3}}{x^{4} \left (x y^{2}+1+x \right ) y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 563
dsolve(diff(y(x),x) = (x*y(x)^2+1)^3/x^4/(x*y(x)^2+1+x)/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {-2 x \left (i x +x +2\right )}}{2 x} \\ y \relax (x ) = \frac {\sqrt {-2 x \left (i x +x +2\right )}}{2 x} \\ y \relax (x ) = -\frac {\sqrt {2}\, \sqrt {x \left (i x -x -2\right )}}{2 x} \\ y \relax (x ) = \frac {\sqrt {2}\, \sqrt {x \left (i x -x -2\right )}}{2 x} \\ \frac {\ln \left (y \relax (x )^{2} x -x +1\right ) y \relax (x )^{2}}{5 y \relax (x )^{2}-5}-\frac {\ln \left (y \relax (x )^{2} x -x +1\right )}{5 \left (y \relax (x )^{2}-1\right )}-\frac {\ln \left (2 y \relax (x )^{4} x^{2}+2 y \relax (x )^{2} x^{2}+4 y \relax (x )^{2} x +x^{2}+2 x +2\right ) y \relax (x )^{4}}{5 \left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right )}-\frac {\ln \left (2 y \relax (x )^{4} x^{2}+2 y \relax (x )^{2} x^{2}+4 y \relax (x )^{2} x +x^{2}+2 x +2\right ) y \relax (x )^{2}}{5 \left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right )}-\frac {\ln \left (2 y \relax (x )^{4} x^{2}+2 y \relax (x )^{2} x^{2}+4 y \relax (x )^{2} x +x^{2}+2 x +2\right )}{10 \left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right )}-\frac {2 \arctan \left (\left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right ) x +2 y \relax (x )^{2}+1\right ) y \relax (x )^{2}}{5}-\frac {\arctan \left (\left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right ) x +2 y \relax (x )^{2}+1\right )}{10}+\frac {4 \arctan \left (\left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right ) x +2 y \relax (x )^{2}+1\right ) y \relax (x )^{6}}{5 \left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right )}+\frac {6 \arctan \left (\left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right ) x +2 y \relax (x )^{2}+1\right ) y \relax (x )^{4}}{5 \left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right )}+\frac {4 \arctan \left (\left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right ) x +2 y \relax (x )^{2}+1\right ) y \relax (x )^{2}}{5 \left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right )}+\frac {\arctan \left (\left (2 y \relax (x )^{4}+2 y \relax (x )^{2}+1\right ) x +2 y \relax (x )^{2}+1\right )}{10 y \relax (x )^{4}+10 y \relax (x )^{2}+5}+\frac {1}{2 x}-\frac {\arctan \left (2 y \relax (x )^{2}+1\right )}{10}+c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.51 (sec). Leaf size: 112
DSolve[y'[x] == (1 + x*y[x]^2)^3/(x^4*y[x]*(1 + x + x*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [2 \left (-\frac {1}{10} \text {ArcTan}\left (2 x y(x)^4+2 x y(x)^2+2 y(x)^2+x+1\right )+\frac {1}{10} \log \left (2 x^2 y(x)^4+2 x^2 y(x)^2+x^2+4 x y(x)^2+2 x+2\right )-\frac {1}{5} \log \left (x y(x)^2-x+1\right )-\frac {1}{2 x}\right )+\frac {1}{5} \text {ArcTan}\left (2 y(x)^2+1\right )=c_1,y(x)\right ] \]