12.7.8 problem 8

Internal problem ID [1718]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Exact equations. Integrating factors. Section 2.6 Page 91
Problem number : 8
Date solved : Monday, January 27, 2025 at 05:32:21 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 27 x y^{2}+8 y^{3}+\left (18 x^{2} y+12 x y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.327 (sec). Leaf size: 33

dsolve((27*x*y(x)^2+8*y(x)^3)+(18*x^2*y(x)+12*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \operatorname {RootOf}\left (4 x^{5} c_1 \,\textit {\_Z}^{15}+9 x^{5} c_1 \,\textit {\_Z}^{10}-1\right )^{5} x \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.145 (sec). Leaf size: 399

DSolve[(27*x*y[x]^2+8*y[x]^3)+(18*x^2*y[x]+12*x*y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 0 \\ y(x)\to \frac {1}{4} \left (\frac {9 x^2}{\sqrt [3]{\frac {-27 x^5+4 \sqrt {e^{6 c_1} \left (-27 x^5+4 e^{6 c_1}\right )}+8 e^{6 c_1}}{x^2}}}+\sqrt [3]{\frac {-27 x^5+4 \sqrt {e^{6 c_1} \left (-27 x^5+4 e^{6 c_1}\right )}+8 e^{6 c_1}}{x^2}}-3 x\right ) \\ y(x)\to \frac {1}{8} \left (-\frac {9 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{\frac {-27 x^5+4 \sqrt {e^{6 c_1} \left (-27 x^5+4 e^{6 c_1}\right )}+8 e^{6 c_1}}{x^2}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {-27 x^5+4 \sqrt {e^{6 c_1} \left (-27 x^5+4 e^{6 c_1}\right )}+8 e^{6 c_1}}{x^2}}-6 x\right ) \\ y(x)\to \frac {1}{8} \left (\frac {9 i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{\frac {-27 x^5+4 \sqrt {e^{6 c_1} \left (-27 x^5+4 e^{6 c_1}\right )}+8 e^{6 c_1}}{x^2}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{\frac {-27 x^5+4 \sqrt {e^{6 c_1} \left (-27 x^5+4 e^{6 c_1}\right )}+8 e^{6 c_1}}{x^2}}-6 x\right ) \\ \end{align*}