Internal problem ID [10115]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 9. Variables
searated or separable. Page 13
Problem number: Ex 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {2 \left (1-y^{2}\right ) x y+\left (x^{2}+1\right ) \left (y^{2}+1\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 75
dsolve(2*(1-y(x)^2)*x*y(x)+(1+x^2)*(1+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x^{2} c_{1}}{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 {x^{2} c_{1}}{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: 7.579 (sec). Leaf size: 98
DSolve[2*(1-y[x]^2)*x*y[x]+(1+x^2)*(1+y[x]^2)*y'[x]==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*}