29.24.6 problem 668

Internal problem ID [5259]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 668
Date solved : Monday, January 27, 2025 at 10:50:08 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 61

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 \left (x \right ) &= \frac {c_{1} x^{2}}{2}+\frac {c_{1}}{2}-\frac {\sqrt {4+\left (x^{2}+1\right )^{2} c_{1}^{2}}}{2} \\ y \left (x \right ) &= \frac {c_{1} x^{2}}{2}+\frac {c_{1}}{2}+\frac {\sqrt {4+\left (x^{2}+1\right )^{2} c_{1}^{2}}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 7.968 (sec). Leaf size: 98

DSolve[(1+x^2)(1+y[x]^2)D[y[x],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*}