Internal problem ID [10139]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 15. Page
22
Problem number: Ex 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {x^{4} y \left (3 y+2 y^{\prime } x \right )+x^{2} \left (4 y+3 y^{\prime } x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 39
dsolve(x^4*y(x)*(3*y(x)+2*x*diff(y(x),x))+ x^2*(4*y(x)+3*x*diff(y(x),x))=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\RootOf \left (x^{2} \textit {\_Z}^{8}-2 \textit {\_Z}^{2} c_{1}-c_{1}\right )^{6} x^{2}-2 c_{1}}{x^{2} c_{1}} \]
✓ Solution by Mathematica
Time used: 60.295 (sec). Leaf size: 1769
DSolve[x^4*y[x]*(3*y[x]+2*x*y'[x])+ x^2*(4*y[x]+3*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*}