Internal problem ID [4182]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter 2, Equations of the first order and degree. page 20
Problem number: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {x y \left (x^{2}+1\right ) y^{\prime }-1-y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 54
dsolve(x*y(x)*(1+x^2)*diff(y(x),x)=1+y(x)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {\left (x^{2}+1\right ) \left (c_{1} x^{2}-1\right )}}{x^{2}+1} \\ y \relax (x ) = -\frac {\sqrt {\left (x^{2}+1\right ) \left (c_{1} x^{2}-1\right )}}{x^{2}+1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.903 (sec). Leaf size: 130
DSolve[x*y[x]*(1+x^2)*y'[x]==1+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-1+\left (-1+e^{2 c_1}\right ) x^2}}{\sqrt {x^2+1}} \\ y(x)\to \frac {\sqrt {-1+\left (-1+e^{2 c_1}\right ) x^2}}{\sqrt {x^2+1}} \\ y(x)\to -i \\ y(x)\to i \\ y(x)\to \frac {\sqrt {x^2+1}}{\sqrt {-x^2-1}} \\ y(x)\to \frac {\sqrt {-x^2-1}}{\sqrt {x^2+1}} \\ \end{align*}