32.6.45 problem Exercise 12.45, page 103

Internal problem ID [5910]
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 12.45, page 103
Date solved : Monday, January 27, 2025 at 01:26:53 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 317

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

\begin{align*} y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}}{4 \sqrt {c_{1}}}-\frac {x^{2} \sqrt {c_{1}}\, \left (i \sqrt {3}-1\right )}{\left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}}{4 \left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 46.009 (sec). Leaf size: 576

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

\begin{align*} y(x)\to \frac {\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+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]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\ y(x)\to \sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}-\frac {x^2}{\sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} \\ y(x)\to \frac {\sqrt [3]{-1} \left (x^2+\sqrt [3]{-1} \left (\sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}\right )}{\sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2+\left (-1-i \sqrt {3}\right ) \left (\sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} \\ \end{align*}