31.1.3 problem 1.3

Internal problem ID [5701]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 2
Problem number : 1.3
Date solved : Monday, January 27, 2025 at 01:08:17 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 54

dsolve(x*y(x)*(1+x^2)*diff(y(x),x)-(1+y(x)^2)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.433 (sec). Leaf size: 131

DSolve[x*y[x]*(1+x^2)*D[y[x],x]-(1+y[x]^2)==0,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*}