29.23.24 problem 655

Internal problem ID [5246]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 655
Date solved : Monday, January 27, 2025 at 10:49:09 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.070 (sec). Leaf size: 238

dsolve(x*(x-3*y(x)^2)*diff(y(x),x)+(2*x-y(x)^2)*y(x) = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {12^{{1}/{3}} \left (x^{3} 12^{{1}/{3}}+{\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{2}/{3}}\right )}{6 x {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-1-i \sqrt {3}\right ) {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x^{3} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )}{12 {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{1}/{3}} x} \\ y \left (x \right ) &= -\frac {2^{{2}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x^{3} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )\right ) 3^{{1}/{3}}}{12 {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{1}/{3}} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 35.350 (sec). Leaf size: 328

DSolve[x(x-3 y[x]^2)D[y[x],x]+(2 x-y[x]^2)y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ \end{align*}