29.24.30 problem 693

Internal problem ID [5283]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 693
Date solved : Monday, January 27, 2025 at 11:03:51 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 279

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

\begin{align*} y \left (x \right ) &= \frac {\left (\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_{1} +18 \ln \left (x \right ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{{2}/{3}}-6 \ln \left (x \right )-6 c_{1} \right ) x}{3 \left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_{1} +18 \ln \left (x \right ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {x \left (\left (\frac {1}{6}+\frac {i \sqrt {3}}{6}\right ) \left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_{1} +18 \ln \left (x \right ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{{2}/{3}}+\left (\ln \left (x \right )+c_{1} \right ) \left (i \sqrt {3}-1\right )\right )}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_{1} +18 \ln \left (x \right ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_{1} +18 \ln \left (x \right ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{{2}/{3}}}{6}+\left (\ln \left (x \right )+c_{1} \right ) \left (1+i \sqrt {3}\right )\right ) x}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_{1} +18 \ln \left (x \right ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 51.721 (sec). Leaf size: 362

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

\begin{align*} y(x)\to \frac {-6^{2/3} x^2 \log (x)+6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}\right ){}^{2/3}}{3 \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}{6^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) x^2 (\log (x)-c_1)}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) x^2 (-\log (x)+c_1)}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}{6^{2/3}} \\ \end{align*}