32.3.3 problem Exact Differential equations. Exercise 9.6, page 79

Internal problem ID [5801]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number : Exact Differential equations. Exercise 9.6, page 79
Date solved : Monday, January 27, 2025 at 01:19:41 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 205

dsolve(2*x*y(x)+(x^2+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}}{4 \sqrt {c_{1}}}-\frac {x^{2} \sqrt {c_{1}}\, \left (i \sqrt {3}-1\right )}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{{2}/{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}}{4 \left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{{1}/{3}} \sqrt {c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 13.199 (sec). Leaf size: 401

DSolve[2*x*y[x]+(x^2+y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2}{4 \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\ y(x)\to 0 \\ y(x)\to \frac {1}{2} \sqrt [6]{x^6} \left (\frac {\left (1-i \sqrt {3}\right ) \left (x^6\right )^{2/3}}{x^4}-i \sqrt {3}-1\right ) \\ y(x)\to \frac {1}{2} \sqrt [6]{x^6} \left (\frac {\left (1+i \sqrt {3}\right ) \left (x^6\right )^{2/3}}{x^4}+i \sqrt {3}-1\right ) \\ y(x)\to \sqrt [6]{x^6}-\frac {\left (x^6\right )^{5/6}}{x^4} \\ \end{align*}