Internal problem ID [13416]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 7. The exact form and general integrating fators. Additional exercises. page
141
Problem number: 7.5 (c).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {\frac {2 y}{x}+\left (4 x^{2} y-3\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 3.234 (sec). Leaf size: 28
dsolve(2*y(x)/x+(4*x^2*y(x)-3)*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\textit {\_Z}^{32} c_{1} -\textit {\_Z}^{24} c_{1} -x^{8}\right )^{8}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 60.256 (sec). Leaf size: 1985
DSolve[2*y[x]/x+(4*x^2*y[x]-3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4 x^2}-\frac {1}{2} \sqrt {\frac {1}{4 x^4}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}+\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}}-\frac {1}{2} \sqrt {\frac {1}{2 x^4}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}-\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}-\frac {1}{4 x^6 \sqrt {\frac {1}{4 x^4}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}+\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}}}} \\ y(x)\to \frac {1}{4 x^2}-\frac {1}{2} \sqrt {\frac {1}{4 x^4}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}+\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}}+\frac {1}{2} \sqrt {\frac {1}{2 x^4}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}-\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}-\frac {1}{4 x^6 \sqrt {\frac {1}{4 x^4}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}+\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}}}} \\ y(x)\to \frac {1}{4 x^2}+\frac {1}{2} \sqrt {\frac {1}{4 x^4}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}+\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}}-\frac {1}{2} \sqrt {\frac {1}{2 x^4}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}-\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}+\frac {1}{4 x^6 \sqrt {\frac {1}{4 x^4}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}+\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}}}} \\ y(x)\to \frac {1}{4 x^2}+\frac {1}{2} \sqrt {\frac {1}{4 x^4}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}+\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}}+\frac {1}{2} \sqrt {\frac {1}{2 x^4}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}-\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}+\frac {1}{4 x^6 \sqrt {\frac {1}{4 x^4}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{-8 c_1} x^2}{\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}+\frac {\sqrt [3]{9 e^{-8 c_1} x^2+\sqrt {3} e^{-24 c_1} \sqrt {e^{24 c_1} x^4 \left (-256 x^8+27 e^{8 c_1}\right )}}}{\sqrt [3]{2} 3^{2/3} x^2}}}} \\ \end{align*}