Internal problem ID [4049]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous
Methods
Problem number: Exercise 12.36, page 103.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {\left (1+x^{2}+y^{2}\right ) y^{\prime }+2 y x +x^{2}+3=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 570
dsolve((x^2+y(x)^2+1)*diff(y(x),x)+2*x*y(x)+x^2+3=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2 \left (x^{2}+1\right )}{\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {x^{2}+1}{\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2}+2}{\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {x^{2}+1}{\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2}+2}{\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+30 x^{4}+6 x^{3} c_{1}+93 x^{2}+54 c_{1} x +9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.665 (sec). Leaf size: 411
DSolve[(x^2+y[x]^2+1)*y'[x]+2*x*y[x]+x^2+3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}{3 \sqrt [3]{2}}-\frac {3 \sqrt [3]{2} \left (x^2+1\right )}{\sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}} \\ y(x)\to \frac {3 \left (1+i \sqrt {3}\right ) \left (x^2+1\right )}{2^{2/3} \sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}{6 \sqrt [3]{2}} \\ y(x)\to \frac {3 \left (1-i \sqrt {3}\right ) \left (x^2+1\right )}{2^{2/3} \sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}{6 \sqrt [3]{2}} \\ \end{align*}