Internal problem ID [13420]
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 (g).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {2 y^{3}+\left (4 y^{3} x^{3}-3 y^{2} x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 32
dsolve(2*y(x)^3+(4*x^3*y(x)^3-3*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (\textit {\_Z}^{32} c_{1} -\textit {\_Z}^{24} c_{1} -x^{8}\right )^{8}}{x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.187 (sec). Leaf size: 1990
DSolve[2*y[x]^3+(4*x^3*y[x]^3-3*x*y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 0 \\ 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*}