34.2.4 problem 4

Internal problem ID [6034]
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
Date solved : Monday, January 27, 2025 at 01:34:00 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.006 (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 &= \frac {\sqrt {\left (x^{2}+1\right ) \left (c_1 \,x^{2}-1\right )}}{x^{2}+1} \\ y &= -\frac {\sqrt {\left (x^{2}+1\right ) \left (c_1 \,x^{2}-1\right )}}{x^{2}+1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.374 (sec). Leaf size: 131

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