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`]]
\[ \boxed {27 x y^{2}+8 y^{3}+\left (18 x^{2} y+12 x y^{2}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.313 (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 \left (x \right ) = 0 \\ y \left (x \right ) = \operatorname {RootOf}\left (4 \textit {\_Z}^{15} c_{1} x^{5}+9 \textit {\_Z}^{10} c_{1} x^{5}-1\right )^{5} x \\ \end{align*}
✓ Solution by Mathematica
Time used: 53.668 (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*}