7.8 problem 8

Internal problem ID [1068]

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.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class A], _rational, [_Abel, 2nd type, class B]]

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

✓ Solution by Maple

Time used: 0.197 (sec). Leaf size: 2051

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 \relax (x ) = 0 \\ y \relax (x ) = \frac {x}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}+\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}} \\ y \relax (x ) = \frac {x}{\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right )^{5} \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}+\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )} \\ y \relax (x ) = \frac {x}{\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right )^{5} \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}+\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )} \\ y \relax (x ) = \frac {x}{\left (\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right )^{5} \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}+\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )} \\ y \relax (x ) = \frac {x}{\left (\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right )^{5} \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}+\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )} \\ y \relax (x ) = \frac {32 x}{-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}-16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )} \\ y \relax (x ) = \frac {32 x}{-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}+16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )} \\ y \relax (x ) = \frac {32 x}{\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right )^{5} \left (-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}-16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )\right )} \\ y \relax (x ) = \frac {32 x}{\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right )^{5} \left (-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}+16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )\right )} \\ y \relax (x ) = \frac {32 x}{\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right )^{5} \left (-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}-16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )\right )} \\ y \relax (x ) = \frac {32 x}{\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right )^{5} \left (-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}+16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )\right )} \\ y \relax (x ) = \frac {32 x}{\left (\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right )^{5} \left (-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}-16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )\right )} \\ y \relax (x ) = \frac {32 x}{\left (\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right )^{5} \left (-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}+16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )\right )} \\ y \relax (x ) = \frac {32 x}{\left (\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right )^{5} \left (-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}-16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )\right )} \\ y \relax (x ) = \frac {32 x}{\left (\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right )^{5} \left (-16 \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {48 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}+16 i \sqrt {3}\, \left (\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}-\frac {3 x^{5} c_{1}}{\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}\right )\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 10.347 (sec). Leaf size: 532

DSolve[(27*x*y[x]^2+8*y[x]^3)+(18*x^2*y[x]+12*x*y[x]^2)*y'[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 {\left (-9-9 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 ) \\ y(x)\to 0 \\ y(x)\to \frac {3 \left (\sqrt [3]{-x^3}+x\right ) \left (-2 x+\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^3}\right )}{8 x} \\ y(x)\to \frac {3 \left (\sqrt [3]{-x^3}+x\right ) \left (-2 x+\left (1-i \sqrt {3}\right ) \sqrt [3]{-x^3}\right )}{8 x} \\ y(x)\to -\frac {3 \left (-\sqrt [3]{-x^3} x+\left (-x^3\right )^{2/3}+x^2\right )}{4 x} \\ \end{align*}