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 (x^{2}+y^{2}+1\right ) y^{\prime }+2 x y+x^{2}+3=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 c_{1}-36 x +4 \sqrt {5 x^{6}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{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}+6 x^{3} c_{1}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.899 (sec). Leaf size: 405
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 {-2 x^2+\sqrt [3]{2} \left (-x \left (x^2+9\right )+\sqrt {93 x^2-6 c_1 \left (x^2+9\right ) x+5 \left (x^2+6\right ) x^4+4+9 c_1{}^2}+3 c_1\right ){}^{2/3}-2}{2^{2/3} \sqrt [3]{-x \left (x^2+9\right )+\sqrt {93 x^2-6 c_1 \left (x^2+9\right ) x+5 \left (x^2+6\right ) x^4+4+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {2 \sqrt [3]{-2} \left (x^2+1\right )+(-2)^{2/3} \left (-x \left (x^2+9\right )+\sqrt {93 x^2-6 c_1 \left (x^2+9\right ) x+5 \left (x^2+6\right ) x^4+4+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2 \sqrt [3]{-x \left (x^2+9\right )+\sqrt {93 x^2-6 c_1 \left (x^2+9\right ) x+5 \left (x^2+6\right ) x^4+4+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {\left (x^2+1\right ) \text {Root}\left [\text {$\#$1}^3+2\&,2\right ]}{\sqrt [3]{-x \left (x^2+9\right )+\sqrt {93 x^2-6 c_1 \left (x^2+9\right ) x+5 \left (x^2+6\right ) x^4+4+9 c_1{}^2}+3 c_1}}-\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-x \left (x^2+9\right )+\sqrt {93 x^2-6 c_1 \left (x^2+9\right ) x+5 \left (x^2+6\right ) x^4+4+9 c_1{}^2}+3 c_1} \\ \end{align*}