Internal problem ID [3729]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1012.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {3 x y^{4} \left (y^{\prime }\right )^{2}-y^{5} y^{\prime }+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.133 (sec). Leaf size: 295
dsolve(3*x*y(x)^4*diff(y(x),x)^2-y(x)^5*diff(y(x),x)+1 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 12^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \relax (x ) = -12^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \relax (x ) = \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) 12^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \relax (x ) = \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) 12^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \relax (x ) = \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) 12^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \relax (x ) = \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) 12^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \relax (x ) = \frac {3^{\frac {1}{6}} \left (-\left (c_{1}^{2}-2 c_{1} x +x^{2}\right ) c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \relax (x ) = -\frac {3^{\frac {1}{6}} \left (-\left (c_{1}^{2}-2 c_{1} x +x^{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 ) 3^{\frac {1}{6}} \left (-\left (c_{1}^{2}-2 c_{1} x +x^{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 ) 3^{\frac {1}{6}} \left (-\left (c_{1}^{2}-2 c_{1} x +x^{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 ) 3^{\frac {1}{6}} \left (-\left (c_{1}^{2}-2 c_{1} x +x^{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 ) 3^{\frac {1}{6}} \left (-\left (c_{1}^{2}-2 c_{1} x +x^{2}\right ) c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.709 (sec). Leaf size: 321
DSolve[3 x y[x]^4 (y'[x])^2 -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]{12 x+e^{c_1}} \\ y(x)\to e^{-\frac {c_1}{6}} \sqrt [3]{6 x+\frac {e^{c_1}}{2}} \\ y(x)\to (-1)^{2/3} e^{-\frac {c_1}{6}} \sqrt [3]{6 x+\frac {e^{c_1}}{2}} \\ y(x)\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (12 x+e^{c_1}\right )} \\ y(x)\to \frac {\sqrt [3]{-e^{-\frac {c_1}{2}} \left (12 x+e^{c_1}\right )}}{\sqrt [3]{2}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (12 x+e^{c_1}\right )}}{\sqrt [3]{2}} \\ y(x)\to \sqrt [6]{x} \text {Root}\left [\text {$\#$1}^6-12\&,3\right ] \\ y(x)\to \sqrt [3]{-2} \sqrt [6]{3} \sqrt [6]{x} \\ y(x)\to \sqrt [6]{x} \text {Root}\left [\text {$\#$1}^6-12\&,1\right ] \\ y(x)\to \sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\ y(x)\to \sqrt [6]{x} \text {Root}\left [\text {$\#$1}^6-12\&,5\right ] \\ y(x)\to \sqrt [6]{x} \text {Root}\left [\text {$\#$1}^6-12\&,4\right ] \\ \end{align*}