Internal problem ID [3407]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 668.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime }+2 x y \left (1-y^{2}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 75
dsolve((x^2+1)*(1+y(x)^2)*diff(y(x),x)+2*x*y(x)*(1-y(x)^2) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {c_{1} x^{2}}{2}+\frac {c_{1}}{2}-\frac {\sqrt {x^{4} c_{1}^{2}+2 x^{2} c_{1}^{2}+c_{1}^{2}+4}}{2} \\ y \relax (x ) = \frac {c_{1} x^{2}}{2}+\frac {c_{1}}{2}+\frac {\sqrt {x^{4} c_{1}^{2}+2 x^{2} c_{1}^{2}+c_{1}^{2}+4}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 7.498 (sec). Leaf size: 98
DSolve[(1+x^2)(1+y[x]^2)y'[x]+2 x y[x](1-y[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (-e^{c_1} \left (x^2+1\right )-\sqrt {4+e^{2 c_1} \left (x^2+1\right )^2}\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {4+e^{2 c_1} \left (x^2+1\right )^2}-e^{c_1} \left (x^2+1\right )\right ) \\ y(x)\to -1 \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}