62.2.3 problem Ex 3

Internal problem ID [12805]
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
Date solved : Tuesday, January 28, 2025 at 04:21:42 AM
CAS classification : [_separable]

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

Solution by Maple

Time used: 0.220 (sec). Leaf size: 61

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 &= \frac {c_{1} x^{2}}{2}+\frac {c_{1}}{2}-\frac {\sqrt {4+\left (x^{2}+1\right )^{2} c_{1}^{2}}}{2} \\ y &= \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: 0.482 (sec). Leaf size: 62

DSolve[2*(1-y[x]^2)*x*y[x]+(1+x^2)*(1+y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]^2+1}{(K[1]-1) K[1] (K[1]+1)}dK[1]\&\right ]\left [\log \left (x^2+1\right )+c_1\right ] \\ y(x)\to -1 \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}