32.4.27 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.19, page 90

Internal problem ID [5838]
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
Date solved : Monday, January 27, 2025 at 01:20:17 PM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 501

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 \left (x \right ) &= \frac {-4 x^{2}-4 a +\left (-4 x^{3}-12 b x -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}}{2 \left (-4 x^{3}-12 b x -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-4 x^{3}-12 b x -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}+\left (x^{2}+a \right ) \left (i \sqrt {3}-1\right )}{\left (-4 x^{3}-12 b x -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 b x -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}}{4}+\left (x^{2}+a \right ) \left (1+i \sqrt {3}\right )}{\left (-4 x^{3}-12 b x -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 7.538 (sec). Leaf size: 396

DSolve[(2*x*y[x]+x^2+b)+(y[x]^2+x^2+a)*D[y[x],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*}