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 : Sunday, March 30, 2025 at 04:09:47 AM
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*}

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

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

\[ c_1 +\frac {x}{{\left (\frac {\left (x^{2}+1\right ) \left (1+y^{2}\right )}{\left (x y-1\right )^{2}}\right )}^{{1}/{4}}}+\frac {\left (x +y\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x +y\right )^{2}}{\left (x y-1\right )^{2}}\right )}{2 x y-2} = 0 \]

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

ode=(1+x^2)D[y[x],x]==1+y[x]^2-2 x y[x](1+y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(y(x)**2 + 1)*y(x) + (x**2 + 1)*Derivative(y(x), x) - y(x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out