27.2.1 problem 1

Internal problem ID [4301]
Book : An introduction to the solution and applications of differential equations, J.W. Searl, 1966
Section : Chapter 4, Ex. 4.2
Problem number : 1
Date solved : Monday, January 27, 2025 at 09:00:38 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 93

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

\begin{align*} y \left (x \right ) &= \frac {-x^{2}-c_{1} x +\sqrt {1+x^{4}+2 c_{1} x^{3}+\left (c_{1}^{2}+2\right ) x^{2}-2 c_{1} x}+1}{2 x} \\ y \left (x \right ) &= \frac {-x^{2}-c_{1} x -\sqrt {1+x^{4}+2 c_{1} x^{3}+\left (c_{1}^{2}+2\right ) x^{2}-2 c_{1} x}+1}{2 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.966 (sec). Leaf size: 95

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

\begin{align*} y(x)\to -\frac {x^2+\sqrt {4 x^2+\left (-x^2+c_1 x+1\right ){}^2}-c_1 x-1}{2 x} \\ y(x)\to \frac {-x^2+\sqrt {4 x^2+\left (-x^2+c_1 x+1\right ){}^2}+c_1 x+1}{2 x} \\ y(x)\to 0 \\ \end{align*}