32.6.36 problem Exercise 12.36, page 103

Internal problem ID [5901]
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
Date solved : Monday, January 27, 2025 at 01:25:23 PM
CAS classification : [_exact, _rational]

\begin{align*} \left (1+x^{2}+y^{2}\right ) y^{\prime }+2 y x +x^{2}+3&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 366

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 \left (x \right ) &= \frac {\left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{{2}/{3}}-4 x^{2}-4}{2 \left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{{2}/{3}}+\left (x^{2}+1\right ) \left (i \sqrt {3}-1\right )}{\left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{{1}/{3}}}{4}+\frac {\left (x^{2}+1\right ) \left (1+i \sqrt {3}\right )}{\left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.625 (sec). Leaf size: 411

DSolve[(x^2+y[x]^2+1)*D[y[x],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*}