Internal problem ID [6126]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 18.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {9 x y^{4} \left (y^{\prime }\right )^{2}-3 y^{5} y^{\prime }-1=0} \end {gather*}
✓ Solution by Maple
Time used: 1.047 (sec). Leaf size: 283
dsolve(9*x*y(x)^4*diff(y(x),x)^2-3*y(x)^5*diff(y(x),x)-1=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{6}} \\ y \relax (x ) = -2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{6}} \\ y \relax (x ) = \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) 2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{6}} \\ y \relax (x ) = \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) 2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{6}} \\ y \relax (x ) = \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) 2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{6}} \\ y \relax (x ) = \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) 2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{6}} \\ y \relax (x ) = \frac {\left (\left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \relax (x ) = -\frac {\left (\left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \relax (x ) = \frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \relax (x ) = \frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \relax (x ) = \frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \relax (x ) = \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.137 (sec). Leaf size: 322
DSolve[9*x*y[x]^4*(y'[x])^2-3*y[x]^5*y'[x]-1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt [3]{-\frac {1}{2}} e^{-\frac {c_1}{6}} \sqrt [3]{-4 x+e^{c_1}} \\ y(x)\to \frac {e^{-\frac {c_1}{6}} \sqrt [3]{-4 x+e^{c_1}}}{\sqrt [3]{2}} \\ y(x)\to \frac {(-1)^{2/3} e^{-\frac {c_1}{6}} \sqrt [3]{-4 x+e^{c_1}}}{\sqrt [3]{2}} \\ y(x)\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (-4 x+e^{c_1}\right )} \\ y(x)\to \frac {\sqrt [3]{e^{-\frac {c_1}{2}} \left (4 x-e^{c_1}\right )}}{\sqrt [3]{2}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (-4 x+e^{c_1}\right )}}{\sqrt [3]{2}} \\ y(x)\to -i \sqrt [3]{2} \sqrt [6]{x} \\ y(x)\to i \sqrt [3]{2} \sqrt [6]{x} \\ y(x)\to -\sqrt [6]{-1} \sqrt [3]{2} \sqrt [6]{x} \\ y(x)\to \sqrt [6]{-1} \sqrt [3]{2} \sqrt [6]{x} \\ y(x)\to -(-1)^{5/6} \sqrt [3]{2} \sqrt [6]{x} \\ y(x)\to (-1)^{5/6} \sqrt [3]{2} \sqrt [6]{x} \\ \end{align*}