60.1.150 problem 151

Internal problem ID [10164]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 151
Date solved : Tuesday, January 28, 2025 at 04:26:06 PM
CAS classification : [_rational, _Abel]

\begin{align*} \left (x^{2}+1\right ) y^{\prime }+\left (1+y^{2}\right ) \left (2 y x -1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 65

dsolve((x^2+1)*diff(y(x),x) + (y(x)^2+1)*(2*x*y(x) - 1)=0,y(x), singsol=all)
 

\[ c_{1} +\frac {x}{{\left (\frac {\left (x^{2}+1\right ) \left (1+y^{2}\right )}{\left (x y-1\right )^{2}}\right )}^{{1}/{4}}}+\frac {\left (x +y\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x +y\right )^{2}}{\left (x y-1\right )^{2}}\right )}{2 x y-2} = 0 \]

✓ Solution by Mathematica

Time used: 0.427 (sec). Leaf size: 203

DSolve[(x^2+1)*D[y[x],x] + (y[x]^2+1)*(2*x*y[x] - 1)==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [c_1=\frac {\frac {1}{2} \left (\frac {1}{\frac {i x}{x^2+1}-\frac {i x^2 y(x)}{x^2+1}}+\frac {i}{x}\right ) \sqrt [4]{1-\left (\frac {1}{\frac {i x}{x^2+1}-\frac {i x^2 y(x)}{x^2+1}}+\frac {i}{x}\right )^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\left (\frac {1}{\frac {i x}{x^2+1}-\frac {i x^2 y(x)}{x^2+1}}+\frac {i}{x}\right )^2\right )+i x}{\sqrt [4]{-1+\left (\frac {1}{\frac {i x}{x^2+1}-\frac {i x^2 y(x)}{x^2+1}}+\frac {i}{x}\right )^2}},y(x)\right ] \]