29.24.34 problem 697

Internal problem ID [5287]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 697
Date solved : Monday, January 27, 2025 at 11:04:26 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x \left (x^{4}-2 y^{3}\right ) y^{\prime }+\left (2 x^{4}+y^{3}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.627 (sec). Leaf size: 28

dsolve(x*(x^4-2*y(x)^3)*diff(y(x),x)+(2*x^4+y(x)^3)*y(x) = 0,y(x), singsol=all)
 

\[ \ln \left (x \right )-c_{1} +\frac {3 \ln \left (\frac {y \left (x \right ) \left (-2 x^{4}+y \left (x \right )^{3}\right )}{x^{{16}/{3}}}\right )}{10} = 0 \]

✓ Solution by Mathematica

Time used: 60.144 (sec). Leaf size: 1139

DSolve[x(x^4-2 y[x]^3)D[y[x],x]+(2 x^4+y[x]^3)y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{6} \left (-\sqrt [6]{2} 3^{2/3} \sqrt {\frac {4 \sqrt [3]{6} c_1 x^2+\left (9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}}}}-3 \sqrt {-\frac {\sqrt [3]{18 x^8-2 \sqrt {81 x^{16}-384 c_1{}^3 x^6}}}{3^{2/3}}-\frac {4\ 2^{2/3} c_1 x^2}{\sqrt [3]{27 x^8-3 \sqrt {81 x^{16}-384 c_1{}^3 x^6}}}-\frac {4 \sqrt {3} x^4}{\sqrt {\frac {4\ 6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}}}}}}\right ) \\ y(x)\to \frac {1}{6} \left (3 \sqrt {-\frac {\sqrt [3]{18 x^8-2 \sqrt {81 x^{16}-384 c_1{}^3 x^6}}}{3^{2/3}}-\frac {4\ 2^{2/3} c_1 x^2}{\sqrt [3]{27 x^8-3 \sqrt {81 x^{16}-384 c_1{}^3 x^6}}}-\frac {4 \sqrt {3} x^4}{\sqrt {\frac {4\ 6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}}}}}}-\sqrt [6]{2} 3^{2/3} \sqrt {\frac {4 \sqrt [3]{6} c_1 x^2+\left (9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}}}}\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt [6]{2} 3^{2/3} \sqrt {\frac {4 \sqrt [3]{6} c_1 x^2+\left (9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}}}}-3 \sqrt {-\frac {\sqrt [3]{18 x^8-2 \sqrt {81 x^{16}-384 c_1{}^3 x^6}}}{3^{2/3}}-\frac {4\ 2^{2/3} c_1 x^2}{\sqrt [3]{27 x^8-3 \sqrt {81 x^{16}-384 c_1{}^3 x^6}}}+\frac {4 \sqrt {3} x^4}{\sqrt {\frac {4\ 6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}}}}}}\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt [6]{2} 3^{2/3} \sqrt {\frac {4 \sqrt [3]{6} c_1 x^2+\left (9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}}}}+3 \sqrt {-\frac {\sqrt [3]{18 x^8-2 \sqrt {81 x^{16}-384 c_1{}^3 x^6}}}{3^{2/3}}-\frac {4\ 2^{2/3} c_1 x^2}{\sqrt [3]{27 x^8-3 \sqrt {81 x^{16}-384 c_1{}^3 x^6}}}+\frac {4 \sqrt {3} x^4}{\sqrt {\frac {4\ 6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 c_1{}^3 x^6}}}}}}\right ) \\ \end{align*}