6.7.8 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 : Tuesday, September 30, 2025 at 05:16:34 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*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 413

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

\begin{align*} y &= 0 \\ y &= \frac {\frac {{\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{1}/{3}}}{x c_1}+\frac {9 x^{3} c_1^{3}}{{\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{1}/{3}}}-3 c_1 x}{4 c_1} \\ y &= \frac {9 i \sqrt {3}\, c_1^{4} x^{4}-9 x^{4} c_1^{4}-6 c_1^{2} x^{2} {\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{1}/{3}}-i \sqrt {3}\, {\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{2}/{3}}-{\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{2}/{3}}}{8 c_1^{2} x {\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{1}/{3}}} \\ y &= -\frac {9 i \sqrt {3}\, c_1^{4} x^{4}+9 x^{4} c_1^{4}+6 c_1^{2} x^{2} {\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{1}/{3}}-i \sqrt {3}\, {\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{2}/{3}}+{\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{2}/{3}}}{8 c_1^{2} x {\left (\left (-27 c_1^{5} x^{5}+4 \sqrt {-27 c_1^{5} x^{5}+4}+8\right ) c_1 x \right )}^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 54.649 (sec). Leaf size: 534

ode=(27*x*y[x]^2+8*y[x]^3)+(18*x^2*y[x]+12*x*y[x]^2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},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 )\\ 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*}

✓ Sympy. Time used: 44.534 (sec). Leaf size: 350

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(27*x*y(x)**2 + (18*x**2*y(x) + 12*x*y(x)**2)*Derivative(y(x), x) + 8*y(x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\frac {18 x^{2}}{\sqrt [3]{- \frac {32 C_{1}}{x^{2}} + 27 x^{3} + 8 \sqrt {C_{1} \left (\frac {16 C_{1}}{x^{4}} - 27 x\right )}}} - 3 x + 3 \sqrt {3} i x - \sqrt [3]{- \frac {32 C_{1}}{x^{2}} + 27 x^{3} + 8 \sqrt {C_{1} \left (\frac {16 C_{1}}{x^{4}} - 27 x\right )}} - \sqrt {3} i \sqrt [3]{- \frac {32 C_{1}}{x^{2}} + 27 x^{3} + 8 \sqrt {C_{1} \left (\frac {16 C_{1}}{x^{4}} - 27 x\right )}}}{4 \left (1 - \sqrt {3} i\right )}, \ y{\left (x \right )} = \frac {\frac {18 x^{2}}{\sqrt [3]{- \frac {32 C_{1}}{x^{2}} + 27 x^{3} + 8 \sqrt {C_{1} \left (\frac {16 C_{1}}{x^{4}} - 27 x\right )}}} - 3 x - 3 \sqrt {3} i x - \sqrt [3]{- \frac {32 C_{1}}{x^{2}} + 27 x^{3} + 8 \sqrt {C_{1} \left (\frac {16 C_{1}}{x^{4}} - 27 x\right )}} + \sqrt {3} i \sqrt [3]{- \frac {32 C_{1}}{x^{2}} + 27 x^{3} + 8 \sqrt {C_{1} \left (\frac {16 C_{1}}{x^{4}} - 27 x\right )}}}{4 \left (1 + \sqrt {3} i\right )}, \ y{\left (x \right )} = - \frac {9 x^{2}}{4 \sqrt [3]{- \frac {32 C_{1}}{x^{2}} + 27 x^{3} + 8 \sqrt {C_{1} \left (\frac {16 C_{1}}{x^{4}} - 27 x\right )}}} - \frac {3 x}{4} - \frac {\sqrt [3]{- \frac {32 C_{1}}{x^{2}} + 27 x^{3} + 8 \sqrt {C_{1} \left (\frac {16 C_{1}}{x^{4}} - 27 x\right )}}}{4}\right ] \]