29.11.10 problem 301

Internal problem ID [4901]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 11
Problem number : 301
Date solved : Tuesday, January 28, 2025 at 02:40:17 PM
CAS classification : [_rational, _Abel]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 65

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

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

✓ Solution by Mathematica

Time used: 0.439 (sec). Leaf size: 203

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

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