Internal problem ID [3986]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number: Recognizable Exact Differential equations. Integrating factors. Exercise
10.19, page 90.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {2 y x +x^{2}+b +\left (a +x^{2}+y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 810
dsolve((2*x*y(x)+x^2+b)+(y(x)^2+x^2+a)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 \left (x^{2}+a \right )}{\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x^{2}+a}{\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2}+2 a}{\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x^{2}+a}{\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2}+2 a}{\left (-4 x^{3}-12 b x -12 c_{1}+4 \sqrt {5 x^{6}+12 a \,x^{4}+6 b \,x^{4}+12 a^{2} x^{2}+9 b^{2} x^{2}+6 x^{3} c_{1}+4 a^{3}+18 b x c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.197 (sec). Leaf size: 396
DSolve[(2*x*y[x]+x^2+b)+(y[x]^2+x^2+a)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1\right ){}^{2/3}-2 a-2 x^2}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}}{2 \sqrt [3]{2}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}}{2 \sqrt [3]{2}} \\ \end{align*}