29.21.10 problem 586

Internal problem ID [5180]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 586
Date solved : Monday, January 27, 2025 at 10:18:02 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 2.017 (sec). Leaf size: 39

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

\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}-2 \textit {\_Z}^{2} c_{1} -c_{1} \right )^{6} x^{2}-2 c_{1}}{x^{2} c_{1}} \]

✓ Solution by Mathematica

Time used: 60.317 (sec). Leaf size: 1769

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

\begin{align*} y(x)\to -\frac {1}{2 x^2}+\frac {\sqrt {\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}+\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}}}{2 \sqrt {3}}-\frac {1}{2} \sqrt {\frac {2}{x^4}+\frac {2\ 2^{2/3} e^{-2 c_1}}{\sqrt [3]{3} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}-\frac {\sqrt [3]{2} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{3^{2/3} x^6}-\frac {2 \sqrt {3}}{x^6 \sqrt {\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}+\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}}}} \\ y(x)\to -\frac {1}{2 x^2}+\frac {\sqrt {\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}+\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}}}{2 \sqrt {3}}+\frac {1}{2} \sqrt {\frac {2}{x^4}+\frac {2\ 2^{2/3} e^{-2 c_1}}{\sqrt [3]{3} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}-\frac {\sqrt [3]{2} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{3^{2/3} x^6}-\frac {2 \sqrt {3}}{x^6 \sqrt {\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}+\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}}}} \\ y(x)\to -\frac {1}{2 x^2}-\frac {\sqrt {\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}+\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}}}{2 \sqrt {3}}-\frac {1}{2} \sqrt {\frac {2}{x^4}+\frac {2\ 2^{2/3} e^{-2 c_1}}{\sqrt [3]{3} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}-\frac {\sqrt [3]{2} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{3^{2/3} x^6}+\frac {2 \sqrt {3}}{x^6 \sqrt {\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}+\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}}}} \\ y(x)\to -\frac {1}{2 x^2}-\frac {\sqrt {\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}+\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}}}{2 \sqrt {3}}+\frac {1}{2} \sqrt {\frac {2}{x^4}+\frac {2\ 2^{2/3} e^{-2 c_1}}{\sqrt [3]{3} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}-\frac {\sqrt [3]{2} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{3^{2/3} x^6}+\frac {2 \sqrt {3}}{x^6 \sqrt {\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}+\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}}}} \\ \end{align*}