Internal problem ID [7731]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 151.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Abel]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime }+\left (y^{2}+1\right ) \left (2 y x -1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 85
dsolve((x^2+1)*diff(y(x),x) + (y(x)^2+1)*(2*x*y(x) - 1)=0,y(x), singsol=all)
\[ c_{1}+\frac {x}{\left (\left (\frac {1}{x}+\frac {x^{2}}{\frac {x^{4} y \relax (x )}{x^{2}+1}-\frac {x^{3}}{x^{2}+1}}\right )^{2}+1\right )^{\frac {1}{4}}}+\frac {\left (y \relax (x )+x \right ) \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (y \relax (x )+x \right )^{2}}{\left (x y \relax (x )-1\right )^{2}}\right )}{2 x y \relax (x )-2} = 0 \]
✓ Solution by Mathematica
Time used: 0.693 (sec). Leaf size: 203
DSolve[(x^2+1)*y'[x] + (y[x]^2+1)*(2*x*y[x] - 1)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {\frac {1}{2} \left (\frac {1}{\frac {i x}{x^2+1}-\frac {i x^2 y(x)}{x^2+1}}+\frac {i}{x}\right ) \sqrt [4]{1-\left (\frac {1}{\frac {i x}{x^2+1}-\frac {i x^2 y(x)}{x^2+1}}+\frac {i}{x}\right )^2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};\left (\frac {1}{\frac {i x}{x^2+1}-\frac {i x^2 y(x)}{x^2+1}}+\frac {i}{x}\right )^2\right )+i x}{\sqrt [4]{-1+\left (\frac {1}{\frac {i x}{x^2+1}-\frac {i x^2 y(x)}{x^2+1}}+\frac {i}{x}\right )^2}},y(x)\right ] \]