Internal problem ID [7254]
Internal file name [OUTPUT/6240_Sunday_June_05_2022_04_33_59_PM_32030808/index.tex]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 29.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "dAlembert"
Maple gives the following as the ode type
[[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {\left (y-2 x y^{\prime }\right )^{2}-{y^{\prime }}^{3}=0} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} \left (-2 x p +y \right )^{2}-p^{3} = 0 \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= 2 x p +p^{\frac {3}{2}}\tag {1A}\\ y &= 2 x p -p^{\frac {3}{2}}\tag {2A} \end {align*}
This has the form \begin {align*} y=xf(p)+g(p)\tag {*} \end {align*}
Where \(f,g\) are functions of \(p=y'(x)\). Each of the above ode’s is dAlembert ode which is now solved. Solving ode 1A Taking derivative of (*) w.r.t. \(x\) gives \begin {align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end {align*}
Comparing the form \(y=x f + g\) to (1A) shows that \begin {align*} f &= 2 p\\ g &= p^{\frac {3}{2}} \end {align*}
Hence (2) becomes \begin {align*} -p = \left (2 x +\frac {3 \sqrt {p}}{2}\right ) p^{\prime }\left (x \right )\tag {2A} \end {align*}
The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives \begin {align*} -p = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=0 \end {align*}
Substituting these in (1A) gives \begin {align*} y&=0 \end {align*}
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in \begin {align*} p^{\prime }\left (x \right ) = -\frac {p \left (x \right )}{2 x +\frac {3 \sqrt {p \left (x \right )}}{2}}\tag {3} \end {align*}
This ODE is now solved for \(p \left (x \right )\).
Inverting the above ode gives \begin {align*} \frac {d}{d p}x \left (p \right ) = -\frac {2 x \left (p \right )+\frac {3 \sqrt {p}}{2}}{p}\tag {4} \end {align*}
This ODE is now solved for \(x \left (p \right )\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d p}x \left (p \right ) + p(p)x \left (p \right ) &= q(p) \end {align*}
Where here \begin {align*} p(p) &=\frac {2}{p}\\ q(p) &=-\frac {3}{2 \sqrt {p}} \end {align*}
Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )+\frac {2 x \left (p \right )}{p} = -\frac {3}{2 \sqrt {p}} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {2}{p}d p} \\ &= p^{2} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}}\left ( \mu x\right ) &= \left (\mu \right ) \left (-\frac {3}{2 \sqrt {p}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left (p^{2} x\right ) &= \left (p^{2}\right ) \left (-\frac {3}{2 \sqrt {p}}\right )\\ \mathrm {d} \left (p^{2} x\right ) &= \left (-\frac {3 p^{\frac {3}{2}}}{2}\right )\, \mathrm {d} p \end {align*}
Integrating gives \begin {align*} p^{2} x &= \int {-\frac {3 p^{\frac {3}{2}}}{2}\,\mathrm {d} p}\\ p^{2} x &= -\frac {3 p^{\frac {5}{2}}}{5} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =p^{2}\) results in \begin {align*} x \left (p \right ) &= -\frac {3 \sqrt {p}}{5}+\frac {c_{1}}{p^{2}} \end {align*}
Now we need to eliminate \(p\) between the above and (1A). One way to do this is by solving (1) for \(p\). This results in \begin {align*} p&={\left (\frac {\left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{6}+\frac {8 x^{2}}{3 \left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}-\frac {2 x}{3}\right )}^{2}\\ p&={\left (-\frac {\left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {4 x^{2}}{3 \left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}-\frac {2 x}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {8 x^{2}}{3 \left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2}\\ p&={\left (-\frac {\left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {4 x^{2}}{3 \left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}-\frac {2 x}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {8 x^{2}}{3 \left (108 y-64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2} \end {align*}
Substituting the above in the solution for \(x\) found above gives \begin{align*} x&=\frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}-128 x^{6}+160 x^{3} y-27 y^{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-2048 x \left (-\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{4}+x \left (-\frac {\left (x^{3}-\frac {15 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{4}+x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}-82944 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \\ x&=\frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}-221184 \left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+3538944 x \left (\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (-i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+16 x^{2}+8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \\ x&=\frac {\left (\left (-82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}+221184 \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right ) \left (1+i \sqrt {3}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-3538944 x \left (-\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-16 x^{2}-8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \\ \end{align*} Solving ode 2A Taking derivative of (*) w.r.t. \(x\) gives \begin {align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end {align*}
Comparing the form \(y=x f + g\) to (1A) shows that \begin {align*} f &= 2 p\\ g &= -p^{\frac {3}{2}} \end {align*}
Hence (2) becomes \begin {align*} -p = \left (2 x -\frac {3 \sqrt {p}}{2}\right ) p^{\prime }\left (x \right )\tag {2A} \end {align*}
The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives \begin {align*} -p = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=0 \end {align*}
Substituting these in (1A) gives \begin {align*} y&=0 \end {align*}
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in \begin {align*} p^{\prime }\left (x \right ) = -\frac {p \left (x \right )}{2 x -\frac {3 \sqrt {p \left (x \right )}}{2}}\tag {3} \end {align*}
This ODE is now solved for \(p \left (x \right )\).
Inverting the above ode gives \begin {align*} \frac {d}{d p}x \left (p \right ) = -\frac {2 x \left (p \right )-\frac {3 \sqrt {p}}{2}}{p}\tag {4} \end {align*}
This ODE is now solved for \(x \left (p \right )\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d p}x \left (p \right ) + p(p)x \left (p \right ) &= q(p) \end {align*}
Where here \begin {align*} p(p) &=\frac {2}{p}\\ q(p) &=\frac {3}{2 \sqrt {p}} \end {align*}
Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )+\frac {2 x \left (p \right )}{p} = \frac {3}{2 \sqrt {p}} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {2}{p}d p} \\ &= p^{2} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}}\left ( \mu x\right ) &= \left (\mu \right ) \left (\frac {3}{2 \sqrt {p}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left (p^{2} x\right ) &= \left (p^{2}\right ) \left (\frac {3}{2 \sqrt {p}}\right )\\ \mathrm {d} \left (p^{2} x\right ) &= \left (\frac {3 p^{\frac {3}{2}}}{2}\right )\, \mathrm {d} p \end {align*}
Integrating gives \begin {align*} p^{2} x &= \int {\frac {3 p^{\frac {3}{2}}}{2}\,\mathrm {d} p}\\ p^{2} x &= \frac {3 p^{\frac {5}{2}}}{5} + c_{2} \end {align*}
Dividing both sides by the integrating factor \(\mu =p^{2}\) results in \begin {align*} x \left (p \right ) &= \frac {3 \sqrt {p}}{5}+\frac {c_{2}}{p^{2}} \end {align*}
Now we need to eliminate \(p\) between the above and (1A). One way to do this is by solving (1) for \(p\). This results in \begin {align*} p&={\left (\frac {\left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{6}+\frac {8 x^{2}}{3 \left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}+\frac {2 x}{3}\right )}^{2}\\ p&={\left (-\frac {\left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {4 x^{2}}{3 \left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}+\frac {2 x}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {8 x^{2}}{3 \left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2}\\ p&={\left (-\frac {\left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {4 x^{2}}{3 \left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}+\frac {2 x}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {8 x^{2}}{3 \left (-108 y+64 x^{3}+12 \sqrt {-96 x^{3} y+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2} \end {align*}
Substituting the above in the solution for \(x\) found above gives \begin{align*} x&=\frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}+128 x^{6}-160 x^{3} y+27 y^{2}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+2048 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{4}+x \left (\frac {\left (x^{3}-\frac {15 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{4}+x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}+82944 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \\ x&=\frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}+221184 \left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-3538944 x \left (-\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}+6635520 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+16 x^{2}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \\ x&=\frac {\frac {27648 \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}\, \left (-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}} \left (3 \left (x^{3}-\frac {3 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}+8 \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right ) \left (1+i \sqrt {3}\right )\right )+128 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right )}{5}+1327104 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{{\left (i \sqrt {3}\, \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-16 i \sqrt {3}\, x^{2}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= 0 \\ \tag{2} x &= \frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}-128 x^{6}+160 x^{3} y-27 y^{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-2048 x \left (-\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{4}+x \left (-\frac {\left (x^{3}-\frac {15 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{4}+x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}-82944 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \\ \tag{3} x &= \frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}-221184 \left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+3538944 x \left (\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (-i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+16 x^{2}+8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \\ \tag{4} x &= \frac {\left (\left (-82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}+221184 \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right ) \left (1+i \sqrt {3}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-3538944 x \left (-\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-16 x^{2}-8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \\ \tag{5} y &= 0 \\ \tag{6} x &= \frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}+128 x^{6}-160 x^{3} y+27 y^{2}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+2048 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{4}+x \left (\frac {\left (x^{3}-\frac {15 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{4}+x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}+82944 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \\ \tag{7} x &= \frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}+221184 \left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-3538944 x \left (-\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}+6635520 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+16 x^{2}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \\ \tag{8} x &= \frac {\frac {27648 \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}\, \left (-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}} \left (3 \left (x^{3}-\frac {3 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}+8 \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right ) \left (1+i \sqrt {3}\right )\right )+128 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right )}{5}+1327104 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{{\left (i \sqrt {3}\, \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-16 i \sqrt {3}\, x^{2}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \\ \end{align*}
Verification of solutions
\[ y = 0 \] Verified OK.
\[ x = \frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}-128 x^{6}+160 x^{3} y-27 y^{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-2048 x \left (-\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{4}+x \left (-\frac {\left (x^{3}-\frac {15 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{4}+x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}-82944 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \] Warning, solution could not be verified
\[ x = \frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}-221184 \left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+3538944 x \left (\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (-i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+16 x^{2}+8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified
\[ x = \frac {\left (\left (-82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}+221184 \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right ) \left (1+i \sqrt {3}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-3538944 x \left (-\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-16 x^{2}-8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified
\[ y = 0 \] Verified OK.
\[ x = \frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}+128 x^{6}-160 x^{3} y+27 y^{2}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+2048 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{4}+x \left (\frac {\left (x^{3}-\frac {15 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{4}+x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}+82944 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \] Warning, solution could not be verified
\[ x = \frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}+221184 \left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-3538944 x \left (-\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}+6635520 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+16 x^{2}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified
\[ x = \frac {\frac {27648 \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}\, \left (-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}} \left (3 \left (x^{3}-\frac {3 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}+8 \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right ) \left (1+i \sqrt {3}\right )\right )+128 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right )}{5}+1327104 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{{\left (i \sqrt {3}\, \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-16 i \sqrt {3}\, x^{2}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y-2 x y^{\prime }\right )^{2}-{y^{\prime }}^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {4 x^{2}}{3}, y^{\prime }=-\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {4 x^{2}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {4 x^{2}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {4 x^{2}}{3} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {4 x^{2}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {4 x^{2}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {4 y x}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y+108 y^{2}+512 x^{6}+12 \sqrt {-96 y^{3} x^{3}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying simple symmetries for implicit equations Successful isolation of dy/dx: 3 solutions were found. Trying to solve each resulting ODE. *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation -> Solving 1st order ODE of high degree, Lie methods, 1st trial `, `-> Computing symmetries using: way = 2 `, `-> Computing symmetries using: way = 2 -> Solving 1st order ODE of high degree, 2nd attempt. Trying parametric methods trying dAlembert <- dAlembert successful <- dAlembert successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 73
dsolve((y(x)-2*x*diff(y(x),x))^2= diff(y(x),x)^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ \left [x \left (\textit {\_T} \right ) &= \frac {3 \textit {\_T}^{\frac {5}{2}}+5 c_{1}}{5 \textit {\_T}^{2}}, y \left (\textit {\_T} \right ) &= \frac {\textit {\_T}^{\frac {5}{2}}+10 c_{1}}{5 \textit {\_T}}\right ] \\ \left [x \left (\textit {\_T} \right ) &= \frac {-3 \textit {\_T}^{\frac {5}{2}}+5 c_{1}}{5 \textit {\_T}^{2}}, y \left (\textit {\_T} \right ) &= \frac {-\textit {\_T}^{\frac {5}{2}}+10 c_{1}}{5 \textit {\_T}}\right ] \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(y[x]-2*x*y'[x])^2== y'[x]^3,y[x],x,IncludeSingularSolutions -> True]
Timed out