Internal problem ID [5346]
Internal file name [OUTPUT/4837_Sunday_February_04_2024_12_46_14_AM_38484777/index.tex]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page
74
Problem number: 18.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "first_order_ode_lie_symmetry_calculated"
Maple gives the following as the ode type
[[_1st_order, _with_linear_symmetries]]
\[ \boxed {{y^{\prime }}^{3}-4 y^{\prime } x^{4}+8 y x^{3}=0} \] Solving the given ode for \(y^{\prime }\) results in \(3\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=2 \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right ) x \tag {1} \\ y^{\prime }&=2 \left (-\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \tag {2} \\ y^{\prime }&=2 \left (-\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \tag {3} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Writing the ode as \begin {align*} y^{\prime }&=\frac {x \left (\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+12 x^{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\\ y^{\prime }&= \omega \left ( x,y\right ) \end {align*}
The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end {align*}
The type of this ode is not in the lookup table. To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives \begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*} Where the unknown coefficients are \[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \] Substituting equations (1E,2E) and \(\omega \) into (A) gives \begin{equation} \tag{5E} b_{2}+\frac {x \left (\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+12 x^{2}\right ) \left (b_{3}-a_{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}-\frac {x^{2} {\left (\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+12 x^{2}\right )}^{2} a_{3}}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}}-\left (\frac {\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+12 x^{2}}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}+\frac {x \left (-\frac {288 x^{5}}{\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \sqrt {-12 x^{6}+81 y^{2}}}+24 x \right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}+\frac {48 x^{6} \left (\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+12 x^{2}\right )}{\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \sqrt {-12 x^{6}+81 y^{2}}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\left (\frac {2 x \left (-108+\frac {972 y}{\sqrt {-12 x^{6}+81 y^{2}}}\right )}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}}-\frac {x \left (\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+12 x^{2}\right ) \left (-108+\frac {972 y}{\sqrt {-12 x^{6}+81 y^{2}}}\right )}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}}}\right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation} Putting the above in normal form gives \[ -\frac {48 \left (-12 x^{6}+81 y^{2}\right )^{\frac {3}{2}} x^{2} a_{3}+46656 x^{3} y^{2} a_{2}-34992 x^{2} y^{3} a_{3}+34992 x^{2} y^{2} a_{1}-3 b_{2} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \sqrt {-12 x^{6}+81 y^{2}}+432 \sqrt {-12 x^{6}+81 y^{2}}\, x^{4} b_{2}+432 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} b_{1}+6912 x^{8} y a_{3}-3888 x^{4} y b_{2}-15552 x^{3} y^{2} b_{3}-3888 x^{3} y b_{1}-144 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{7} a_{2}-144 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} a_{1}+\sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} a_{1}+8 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} x^{4} a_{3}+48 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} a_{3}-144 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} y a_{3}+2 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} x a_{2}-\sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} x b_{3}+\sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} y a_{3}-36 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{2} b_{2}+1728 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} y b_{3}+324 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{2} y b_{2}+324 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x \,y^{2} b_{3}-36 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x b_{1}+324 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x y b_{1}-5184 x^{9} a_{2}-3456 x^{8} a_{1}+1728 x^{9} b_{3}-36 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x y b_{3}-5184 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} a_{2} y -3888 \sqrt {-12 x^{6}+81 y^{2}}\, x^{2} a_{1} y}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \sqrt {-12 x^{6}+81 y^{2}}} = 0 \] Setting the numerator to zero gives \begin{equation} \tag{6E} -48 \left (-12 x^{6}+81 y^{2}\right )^{\frac {3}{2}} x^{2} a_{3}-46656 x^{3} y^{2} a_{2}+34992 x^{2} y^{3} a_{3}-34992 x^{2} y^{2} a_{1}+3 b_{2} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \sqrt {-12 x^{6}+81 y^{2}}-432 \sqrt {-12 x^{6}+81 y^{2}}\, x^{4} b_{2}-432 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} b_{1}-6912 x^{8} y a_{3}+3888 x^{4} y b_{2}+15552 x^{3} y^{2} b_{3}+3888 x^{3} y b_{1}+144 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{7} a_{2}+144 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} a_{1}-\sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} a_{1}-8 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} x^{4} a_{3}-48 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} a_{3}+144 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} y a_{3}-2 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} x a_{2}+\sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} x b_{3}-\sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} y a_{3}+36 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{2} b_{2}-1728 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} y b_{3}-324 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{2} y b_{2}-324 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x \,y^{2} b_{3}+36 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x b_{1}-324 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x y b_{1}+5184 x^{9} a_{2}+3456 x^{8} a_{1}-1728 x^{9} b_{3}+36 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x y b_{3}+5184 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} a_{2} y +3888 \sqrt {-12 x^{6}+81 y^{2}}\, x^{2} a_{1} y = 0 \end{equation} Simplifying the above gives \begin{equation} \tag{6E} 9 b_{2} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \sqrt {-12 x^{6}+81 y^{2}}-1296 \sqrt {-12 x^{6}+81 y^{2}}\, x^{4} b_{2}-1296 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} b_{1}-5184 x^{8} y a_{3}+11664 x^{4} y b_{2}+11664 x^{3} y^{2} b_{3}+11664 x^{3} y b_{1}+432 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{7} a_{2}+432 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} a_{1}-3 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} a_{1}-\sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{2} x^{2} a_{3}-24 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} x^{4} a_{3}-144 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} a_{3}+432 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} y a_{3}-6 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} x a_{2}+3 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} x b_{3}-3 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {5}{3}} y a_{3}-144 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right ) x^{3} a_{2}+36 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right ) x^{3} b_{3}-108 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right ) x^{2} a_{1}+108 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{2} b_{2}-1296 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} y b_{3}-972 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{2} y b_{2}-972 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x \,y^{2} b_{3}+108 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x b_{1}-972 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x y b_{1}-5184 x^{9} a_{2}-5184 x^{8} a_{1}-108 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right ) x^{2} y a_{3}+108 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x y b_{3} = 0 \end{equation} Since the PDE has radicals, simplifying gives \[ -2916 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} y^{2} a_{1}+8748 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} y^{2} b_{2}-139968 x^{3} y^{2} a_{2}+104976 x^{2} y^{3} a_{3}-104976 x^{2} y^{2} a_{1}+3456 x^{10} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} a_{3}-432 x^{7} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} b_{3}-1296 x^{6} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} b_{2}-1296 \sqrt {-12 x^{6}+81 y^{2}}\, x^{4} b_{2}-1296 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} b_{1}-11664 x^{2} \sqrt {-12 x^{6}+81 y^{2}}\, y^{2} a_{3}+324 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} y^{2} a_{3}-20736 x^{8} y a_{3}+11664 x^{4} y b_{2}+46656 x^{3} y^{2} b_{3}+11664 x^{3} y b_{1}+1296 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{7} a_{2}+864 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} a_{1}-144 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} a_{3}+864 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{6} y a_{3}+108 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{2} b_{2}-5184 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} y b_{3}-972 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x^{2} y b_{2}+1944 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x \,y^{2} b_{3}+108 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x b_{1}-972 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x y b_{1}-2916 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} y^{3} a_{3}+1728 x^{8} \sqrt {-12 x^{6}+81 y^{2}}\, a_{3}+324 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} a_{1} y -972 b_{2} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \sqrt {-12 x^{6}+81 y^{2}}\, y -23328 x^{4} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} y^{2} a_{3}+15552 x^{9} a_{2}+10368 x^{8} a_{1}-5184 x^{9} b_{3}+2592 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} x^{4} a_{3} y -5832 x \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} y^{2} a_{2}-216 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x y b_{3}+15552 \sqrt {-12 x^{6}+81 y^{2}}\, x^{3} a_{2} y +11664 \sqrt {-12 x^{6}+81 y^{2}}\, x^{2} a_{1} y +648 \sqrt {-12 x^{6}+81 y^{2}}\, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} x a_{2} y = 0 \] Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them. \[ \left \{x, y, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}, \sqrt {-12 x^{6}+81 y^{2}}\right \} \] The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them \[ \left \{x = v_{1}, y = v_{2}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} = v_{3}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} = v_{4}, \sqrt {-12 x^{6}+81 y^{2}} = v_{5}\right \} \] The above PDE (6E) now becomes \begin{equation} \tag{7E} 3456 v_{1}^{10} v_{3} a_{3}+15552 v_{1}^{9} a_{2}-20736 v_{1}^{8} v_{2} a_{3}+1728 v_{1}^{8} v_{5} a_{3}-5184 v_{1}^{9} b_{3}+10368 v_{1}^{8} a_{1}+1296 v_{4} v_{1}^{7} a_{2}+864 v_{4} v_{1}^{6} v_{2} a_{3}-144 v_{5} v_{4} v_{1}^{6} a_{3}-432 v_{1}^{7} v_{4} b_{3}+864 v_{4} v_{1}^{6} a_{1}-23328 v_{1}^{4} v_{3} v_{2}^{2} a_{3}+2592 v_{5} v_{3} v_{1}^{4} a_{3} v_{2}-1296 v_{1}^{6} v_{3} b_{2}-139968 v_{1}^{3} v_{2}^{2} a_{2}+15552 v_{5} v_{1}^{3} a_{2} v_{2}+104976 v_{1}^{2} v_{2}^{3} a_{3}-11664 v_{1}^{2} v_{5} v_{2}^{2} a_{3}+11664 v_{1}^{4} v_{2} b_{2}-1296 v_{5} v_{1}^{4} b_{2}+46656 v_{1}^{3} v_{2}^{2} b_{3}-5184 v_{5} v_{1}^{3} v_{2} b_{3}-104976 v_{1}^{2} v_{2}^{2} a_{1}+11664 v_{5} v_{1}^{2} a_{1} v_{2}-5832 v_{1} v_{4} v_{2}^{2} a_{2}+648 v_{5} v_{4} v_{1} a_{2} v_{2}-2916 v_{4} v_{2}^{3} a_{3}+324 v_{5} v_{4} v_{2}^{2} a_{3}+11664 v_{1}^{3} v_{2} b_{1}-1296 v_{5} v_{1}^{3} b_{1}-972 v_{4} v_{1}^{2} v_{2} b_{2}+108 v_{5} v_{4} v_{1}^{2} b_{2}+1944 v_{4} v_{1} v_{2}^{2} b_{3}-216 v_{5} v_{4} v_{1} v_{2} b_{3}-2916 v_{4} v_{2}^{2} a_{1}+324 v_{5} v_{4} a_{1} v_{2}-972 v_{4} v_{1} v_{2} b_{1}+108 v_{5} v_{4} v_{1} b_{1}+8748 v_{3} v_{2}^{2} b_{2}-972 b_{2} v_{3} v_{5} v_{2} = 0 \end{equation} Collecting the above on the terms \(v_i\) introduced, and these are \[ \{v_{1}, v_{2}, v_{3}, v_{4}, v_{5}\} \] Equation (7E) now becomes \begin{equation} \tag{8E} \left (15552 a_{2}-5184 b_{3}\right ) v_{1}^{9}-1296 v_{5} v_{1}^{3} b_{1}-20736 v_{1}^{8} v_{2} a_{3}+11664 v_{1}^{4} v_{2} b_{2}+11664 v_{1}^{3} v_{2} b_{1}+864 v_{4} v_{1}^{6} a_{1}-2916 v_{4} v_{2}^{3} a_{3}+1728 v_{1}^{8} v_{5} a_{3}+10368 v_{1}^{8} a_{1}+\left (648 a_{2}-216 b_{3}\right ) v_{1} v_{2} v_{4} v_{5}-2916 v_{4} v_{2}^{2} a_{1}+8748 v_{3} v_{2}^{2} b_{2}+104976 v_{1}^{2} v_{2}^{3} a_{3}-104976 v_{1}^{2} v_{2}^{2} a_{1}+3456 v_{1}^{10} v_{3} a_{3}-1296 v_{1}^{6} v_{3} b_{2}-1296 v_{5} v_{1}^{4} b_{2}+2592 v_{5} v_{3} v_{1}^{4} a_{3} v_{2}-11664 v_{1}^{2} v_{5} v_{2}^{2} a_{3}+324 v_{5} v_{4} v_{2}^{2} a_{3}-144 v_{5} v_{4} v_{1}^{6} a_{3}+864 v_{4} v_{1}^{6} v_{2} a_{3}+108 v_{5} v_{4} v_{1}^{2} b_{2}-972 v_{4} v_{1}^{2} v_{2} b_{2}+108 v_{5} v_{4} v_{1} b_{1}-972 v_{4} v_{1} v_{2} b_{1}+324 v_{5} v_{4} a_{1} v_{2}-972 b_{2} v_{3} v_{5} v_{2}-23328 v_{1}^{4} v_{3} v_{2}^{2} a_{3}+11664 v_{5} v_{1}^{2} a_{1} v_{2}+\left (1296 a_{2}-432 b_{3}\right ) v_{1}^{7} v_{4}+\left (-139968 a_{2}+46656 b_{3}\right ) v_{1}^{3} v_{2}^{2}+\left (15552 a_{2}-5184 b_{3}\right ) v_{1}^{3} v_{2} v_{5}+\left (-5832 a_{2}+1944 b_{3}\right ) v_{1} v_{2}^{2} v_{4} = 0 \end{equation} Setting each coefficients in (8E) to zero gives the following equations to solve \begin {align*} -104976 a_{1}&=0\\ -2916 a_{1}&=0\\ 324 a_{1}&=0\\ 864 a_{1}&=0\\ 10368 a_{1}&=0\\ 11664 a_{1}&=0\\ -23328 a_{3}&=0\\ -20736 a_{3}&=0\\ -11664 a_{3}&=0\\ -2916 a_{3}&=0\\ -144 a_{3}&=0\\ 324 a_{3}&=0\\ 864 a_{3}&=0\\ 1728 a_{3}&=0\\ 2592 a_{3}&=0\\ 3456 a_{3}&=0\\ 104976 a_{3}&=0\\ -1296 b_{1}&=0\\ -972 b_{1}&=0\\ 108 b_{1}&=0\\ 11664 b_{1}&=0\\ -1296 b_{2}&=0\\ -972 b_{2}&=0\\ 108 b_{2}&=0\\ 8748 b_{2}&=0\\ 11664 b_{2}&=0\\ -139968 a_{2}+46656 b_{3}&=0\\ -5832 a_{2}+1944 b_{3}&=0\\ 648 a_{2}-216 b_{3}&=0\\ 1296 a_{2}-432 b_{3}&=0\\ 15552 a_{2}-5184 b_{3}&=0 \end {align*}
Solving the above equations for the unknowns gives \begin {align*} a_{1}&=0\\ a_{2}&=a_{2}\\ a_{3}&=0\\ b_{1}&=0\\ b_{2}&=0\\ b_{3}&=3 a_{2} \end {align*}
Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives \begin{align*} \xi &= x \\ \eta &= 3 y \\ \end{align*} The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.
The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}
The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Therefore \begin {align*} \frac {dy}{dx} &= \frac {\eta }{\xi }\\ &= \frac {3 y}{x}\\ &= \frac {3 y}{x} \end {align*}
This is easily solved to give \begin {align*} y = c_{1} x^{3} \end {align*}
Where now the coordinate \(R\) is taken as the constant of integration. Hence \begin {align*} R &= \frac {y}{x^{3}} \end {align*}
And \(S\) is found from \begin {align*} dS &= \frac {dx}{\xi } \\ &= \frac {dx}{x} \end {align*}
Integrating gives \begin {align*} S &= \int { \frac {dx}{T}}\\ &= \ln \left (x \right ) \end {align*}
Where the constant of integration is set to zero as we just need one solution. Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by evaluating \begin {align*} \frac {dS}{dR} &= \frac { S_{x} + \omega (x,y) S_{y} }{ R_{x} + \omega (x,y) R_{y} }\tag {2} \end {align*}
Where in the above \(R_{x},R_{y},S_{x},S_{y}\) are all partial derivatives and \(\omega (x,y)\) is the right hand side of the original ode given by \begin {align*} \omega (x,y) &= \frac {x \left (\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+12 x^{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}} \end {align*}
Evaluating all the partial derivatives gives \begin {align*} R_{x} &= -\frac {3 y}{x^{4}}\\ R_{y} &= \frac {1}{x^{3}}\\ S_{x} &= \frac {1}{x}\\ S_{y} &= 0 \end {align*}
Substituting all the above in (2) and simplifying gives the ode in canonical coordinates. \begin {align*} \frac {dS}{dR} &= \frac {3 x^{3} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{x^{2} \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+12 x^{4}-9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} y}\tag {2A} \end {align*}
We now need to express the RHS as function of \(R\) only. This is done by solving for \(x,y\) in terms of \(R,S\) from the result obtained earlier and simplifying. This gives \begin {align*} \frac {dS}{dR} &= \frac {3 \,12^{\frac {1}{3}} \left (\sqrt {27 R^{2}-4}\, \sqrt {3}-9 R \right )^{\frac {1}{3}}}{12^{\frac {2}{3}} \left (\sqrt {27 R^{2}-4}\, \sqrt {3}-9 R \right )^{\frac {2}{3}}-9 \,12^{\frac {1}{3}} \left (\sqrt {27 R^{2}-4}\, \sqrt {3}-9 R \right )^{\frac {1}{3}} R +12} \end {align*}
The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates \(R,S\). Integrating the above gives \begin {align*} S \left (R \right ) = \int \frac {3 \left (12 \sqrt {81 R^{2}-12}-108 R \right )^{\frac {1}{3}}}{2 \,18^{\frac {1}{3}} {\left (\left (\sqrt {81 R^{2}-12}-9 R \right )^{2}\right )}^{\frac {1}{3}}-9 R \left (12 \sqrt {81 R^{2}-12}-108 R \right )^{\frac {1}{3}}+12}d R +c_{1}\tag {4} \end {align*}
To complete the solution, we just need to transform (4) back to \(x,y\) coordinates. This results in \begin {align*} \ln \left (x \right ) = \int _{}^{\frac {y}{x^{3}}}\frac {3 \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}}{2 \,18^{\frac {1}{3}} {\left (\left (\sqrt {81 \textit {\_a}^{2}-12}-9 \textit {\_a} \right )^{2}\right )}^{\frac {1}{3}}-9 \textit {\_a} \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}+12}d \textit {\_a} +c_{1} \end {align*}
Which simplifies to \begin {align*} \ln \left (x \right ) = \int _{}^{\frac {y}{x^{3}}}\frac {3 \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}}{2 \,18^{\frac {1}{3}} {\left (\left (\sqrt {81 \textit {\_a}^{2}-12}-9 \textit {\_a} \right )^{2}\right )}^{\frac {1}{3}}-9 \textit {\_a} \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}+12}d \textit {\_a} +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \ln \left (x \right ) &= \int _{}^{\frac {y}{x^{3}}}\frac {3 \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}}{2 \,18^{\frac {1}{3}} {\left (\left (\sqrt {81 \textit {\_a}^{2}-12}-9 \textit {\_a} \right )^{2}\right )}^{\frac {1}{3}}-9 \textit {\_a} \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}+12}d \textit {\_a} +c_{1} \\ \end{align*}
Verification of solutions
\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x^{3}}}\frac {3 \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}}{2 \,18^{\frac {1}{3}} {\left (\left (\sqrt {81 \textit {\_a}^{2}-12}-9 \textit {\_a} \right )^{2}\right )}^{\frac {1}{3}}-9 \textit {\_a} \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}+12}d \textit {\_a} +c_{1} \] Verified OK.
Solving equation (2)
Writing the ode as \begin {align*} y^{\prime }&=-\frac {2 x \left (6 i \sqrt {3}\, x^{2}+\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}-6 x^{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}\\ y^{\prime }&= \omega \left ( x,y\right ) \end {align*}
The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end {align*}
The type of this ode is not in the lookup table. To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives \begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*} Where the unknown coefficients are \[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \] Substituting equations (1E,2E) and \(\omega \) into (A) gives \begin{equation} \tag{5E} b_{2}-\frac {2 x \left (6 i \sqrt {3}\, x^{2}+\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}-6 x^{2}\right ) \left (b_{3}-a_{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}-\frac {4 x^{2} \left (6 i \sqrt {3}\, x^{2}+\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}-6 x^{2}\right )^{2} a_{3}}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}-\left (-\frac {2 \left (6 i \sqrt {3}\, x^{2}+\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}-6 x^{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}-\frac {2 x \left (12 i \sqrt {3}\, x -\frac {288 x^{5}}{\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \sqrt {-12 x^{6}+81 y^{2}}}-12 x \right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}-\frac {96 x^{6} \left (6 i \sqrt {3}\, x^{2}+\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}-6 x^{2}\right )}{\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right ) \sqrt {-12 x^{6}+81 y^{2}}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\left (-\frac {4 x \left (-108+\frac {972 y}{\sqrt {-12 x^{6}+81 y^{2}}}\right )}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}+\frac {2 x \left (6 i \sqrt {3}\, x^{2}+\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}-6 x^{2}\right ) \left (-108+\frac {972 y}{\sqrt {-12 x^{6}+81 y^{2}}}\right )}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}\right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation} Putting the above in normal form gives \[ \text {Expression too large to display} \] Setting the numerator to zero gives \begin{equation} \tag{6E} \text {Expression too large to display} \end{equation} Simplifying the above gives \begin{equation} \tag{6E} \text {Expression too large to display} \end{equation} Since the PDE has radicals, simplifying gives \[ \text {Expression too large to display} \] Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them. \[ \left \{x, y, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}, \sqrt {-12 x^{6}+81 y^{2}}\right \} \] The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them \[ \left \{x = v_{1}, y = v_{2}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} = v_{3}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} = v_{4}, \sqrt {-12 x^{6}+81 y^{2}} = v_{5}\right \} \] The above PDE (6E) now becomes \begin{equation} \tag{7E} 46656 v_{5} v_{1}^{2} a_{1} v_{2}+288 v_{5} v_{4} v_{1}^{6} a_{3}-20736 v_{5} v_{1}^{3} v_{2} b_{3}-1728 v_{4} v_{1}^{6} v_{2} a_{3}-216 v_{5} v_{4} v_{1}^{2} b_{2}+1944 v_{4} v_{1}^{2} v_{2} b_{2}-3888 v_{4} v_{1} v_{2}^{2} b_{3}-216 v_{5} v_{4} v_{1} b_{1}+1944 v_{4} v_{1} v_{2} b_{1}+46656 v_{1}^{4} v_{3} v_{2}^{2} a_{3}+11664 v_{1} v_{4} v_{2}^{2} a_{2}+1944 v_{5} v_{3} b_{2} v_{2}-648 v_{5} v_{4} a_{1} v_{2}-46656 v_{1}^{2} v_{5} v_{2}^{2} a_{3}-648 v_{5} v_{4} v_{2}^{2} a_{3}-20736 v_{1}^{9} b_{3}+62208 v_{1}^{9} a_{2}+41472 v_{1}^{8} a_{1}-1944 i \sqrt {3}\, v_{5} v_{3} b_{2} v_{2}-648 i \sqrt {3}\, v_{5} v_{4} v_{2}^{2} a_{3}-3888 i \sqrt {3}\, v_{4} v_{1} v_{2}^{2} b_{3}-1728 i \sqrt {3}\, v_{4} v_{1}^{6} v_{2} a_{3}-46656 i \sqrt {3}\, v_{1}^{4} v_{3} v_{2}^{2} a_{3}+11664 i \sqrt {3}\, v_{1} v_{4} v_{2}^{2} a_{2}+288 i \sqrt {3}\, v_{5} v_{4} v_{1}^{6} a_{3}-1296 v_{5} v_{4} v_{1} a_{2} v_{2}-5184 v_{5} v_{3} v_{1}^{4} a_{3} v_{2}+432 v_{5} v_{4} v_{1} v_{2} b_{3}+6912 i \sqrt {3}\, v_{1}^{10} v_{3} a_{3}-2592 i \sqrt {3}\, v_{4} v_{1}^{7} a_{2}+864 i \sqrt {3}\, v_{1}^{7} v_{4} b_{3}-1728 i \sqrt {3}\, v_{4} v_{1}^{6} a_{1}-2592 i \sqrt {3}\, v_{1}^{6} v_{3} b_{2}+5832 i \sqrt {3}\, v_{4} v_{2}^{3} a_{3}+5832 i \sqrt {3}\, v_{4} v_{2}^{2} a_{1}+17496 i \sqrt {3}\, v_{3} v_{2}^{2} b_{2}-216 i \sqrt {3}\, v_{5} v_{4} v_{1}^{2} b_{2}+1944 i \sqrt {3}\, v_{4} v_{1}^{2} v_{2} b_{2}-216 i \sqrt {3}\, v_{5} v_{4} v_{1} b_{1}+1944 i \sqrt {3}\, v_{4} v_{1} v_{2} b_{1}-648 i \sqrt {3}\, v_{5} v_{4} a_{1} v_{2}+62208 v_{5} v_{1}^{3} a_{2} v_{2}-1296 i \sqrt {3}\, v_{5} v_{4} v_{1} a_{2} v_{2}+432 i \sqrt {3}\, v_{5} v_{4} v_{1} v_{2} b_{3}+5184 i \sqrt {3}\, v_{5} v_{3} v_{1}^{4} a_{3} v_{2}-82944 v_{1}^{8} v_{2} a_{3}+46656 v_{1}^{4} v_{2} b_{2}+186624 v_{1}^{3} v_{2}^{2} b_{3}+46656 v_{1}^{3} v_{2} b_{1}-5184 v_{5} v_{1}^{4} b_{2}-2592 v_{4} v_{1}^{7} a_{2}-5184 v_{5} v_{1}^{3} b_{1}-1728 v_{4} v_{1}^{6} a_{1}+6912 v_{1}^{8} v_{5} a_{3}-6912 v_{1}^{10} v_{3} a_{3}+864 v_{1}^{7} v_{4} b_{3}+2592 v_{1}^{6} v_{3} b_{2}+5832 v_{4} v_{2}^{3} a_{3}+5832 v_{4} v_{2}^{2} a_{1}-17496 v_{3} v_{2}^{2} b_{2}-559872 v_{1}^{3} v_{2}^{2} a_{2}+419904 v_{1}^{2} v_{2}^{3} a_{3}-419904 v_{1}^{2} v_{2}^{2} a_{1} = 0 \end{equation} Collecting the above on the terms \(v_i\) introduced, and these are \[ \{v_{1}, v_{2}, v_{3}, v_{4}, v_{5}\} \] Equation (7E) now becomes \begin{equation} \tag{8E} \left (288 i \sqrt {3}\, a_{3}+288 a_{3}\right ) v_{1}^{6} v_{4} v_{5}+\left (-46656 i \sqrt {3}\, a_{3}+46656 a_{3}\right ) v_{1}^{4} v_{2}^{2} v_{3}+\left (1944 i \sqrt {3}\, b_{2}+1944 b_{2}\right ) v_{1}^{2} v_{2} v_{4}+\left (-216 i \sqrt {3}\, b_{2}-216 b_{2}\right ) v_{1}^{2} v_{4} v_{5}+\left (11664 i \sqrt {3}\, a_{2}-3888 i \sqrt {3}\, b_{3}+11664 a_{2}-3888 b_{3}\right ) v_{1} v_{2}^{2} v_{4}+\left (1944 i \sqrt {3}\, b_{1}+1944 b_{1}\right ) v_{1} v_{2} v_{4}+\left (-216 i \sqrt {3}\, b_{1}-216 b_{1}\right ) v_{1} v_{4} v_{5}+\left (-648 i \sqrt {3}\, a_{3}-648 a_{3}\right ) v_{2}^{2} v_{4} v_{5}+\left (-1944 i \sqrt {3}\, b_{2}+1944 b_{2}\right ) v_{2} v_{3} v_{5}+\left (-648 i \sqrt {3}\, a_{1}-648 a_{1}\right ) v_{2} v_{4} v_{5}+\left (-1728 i \sqrt {3}\, a_{3}-1728 a_{3}\right ) v_{1}^{6} v_{2} v_{4}+46656 v_{5} v_{1}^{2} a_{1} v_{2}-46656 v_{1}^{2} v_{5} v_{2}^{2} a_{3}+41472 v_{1}^{8} a_{1}+\left (62208 a_{2}-20736 b_{3}\right ) v_{1}^{9}+\left (-559872 a_{2}+186624 b_{3}\right ) v_{1}^{3} v_{2}^{2}+\left (6912 i \sqrt {3}\, a_{3}-6912 a_{3}\right ) v_{1}^{10} v_{3}+\left (-2592 i \sqrt {3}\, a_{2}+864 i \sqrt {3}\, b_{3}-2592 a_{2}+864 b_{3}\right ) v_{1}^{7} v_{4}+\left (-2592 i \sqrt {3}\, b_{2}+2592 b_{2}\right ) v_{1}^{6} v_{3}+\left (-1728 i \sqrt {3}\, a_{1}-1728 a_{1}\right ) v_{1}^{6} v_{4}+\left (5832 i \sqrt {3}\, a_{3}+5832 a_{3}\right ) v_{2}^{3} v_{4}+\left (17496 i \sqrt {3}\, b_{2}-17496 b_{2}\right ) v_{2}^{2} v_{3}+\left (5832 i \sqrt {3}\, a_{1}+5832 a_{1}\right ) v_{2}^{2} v_{4}+\left (5184 i \sqrt {3}\, a_{3}-5184 a_{3}\right ) v_{1}^{4} v_{2} v_{3} v_{5}+\left (-1296 i \sqrt {3}\, a_{2}+432 i \sqrt {3}\, b_{3}-1296 a_{2}+432 b_{3}\right ) v_{1} v_{2} v_{4} v_{5}+\left (62208 a_{2}-20736 b_{3}\right ) v_{1}^{3} v_{2} v_{5}-82944 v_{1}^{8} v_{2} a_{3}+46656 v_{1}^{4} v_{2} b_{2}+46656 v_{1}^{3} v_{2} b_{1}-5184 v_{5} v_{1}^{4} b_{2}-5184 v_{5} v_{1}^{3} b_{1}+6912 v_{1}^{8} v_{5} a_{3}+419904 v_{1}^{2} v_{2}^{3} a_{3}-419904 v_{1}^{2} v_{2}^{2} a_{1} = 0 \end{equation} Setting each coefficients in (8E) to zero gives the following equations to solve \begin {align*} -419904 a_{1}&=0\\ 41472 a_{1}&=0\\ 46656 a_{1}&=0\\ -82944 a_{3}&=0\\ -46656 a_{3}&=0\\ 6912 a_{3}&=0\\ 419904 a_{3}&=0\\ -5184 b_{1}&=0\\ 46656 b_{1}&=0\\ -5184 b_{2}&=0\\ 46656 b_{2}&=0\\ -559872 a_{2}+186624 b_{3}&=0\\ 62208 a_{2}-20736 b_{3}&=0\\ -46656 i \sqrt {3}\, a_{3}+46656 a_{3}&=0\\ -2592 i \sqrt {3}\, b_{2}+2592 b_{2}&=0\\ -1944 i \sqrt {3}\, b_{2}+1944 b_{2}&=0\\ -1728 i \sqrt {3}\, a_{1}-1728 a_{1}&=0\\ -1728 i \sqrt {3}\, a_{3}-1728 a_{3}&=0\\ -648 i \sqrt {3}\, a_{1}-648 a_{1}&=0\\ -648 i \sqrt {3}\, a_{3}-648 a_{3}&=0\\ -216 i \sqrt {3}\, b_{1}-216 b_{1}&=0\\ -216 i \sqrt {3}\, b_{2}-216 b_{2}&=0\\ 288 i \sqrt {3}\, a_{3}+288 a_{3}&=0\\ 1944 i \sqrt {3}\, b_{1}+1944 b_{1}&=0\\ 1944 i \sqrt {3}\, b_{2}+1944 b_{2}&=0\\ 5184 i \sqrt {3}\, a_{3}-5184 a_{3}&=0\\ 5832 i \sqrt {3}\, a_{1}+5832 a_{1}&=0\\ 5832 i \sqrt {3}\, a_{3}+5832 a_{3}&=0\\ 6912 i \sqrt {3}\, a_{3}-6912 a_{3}&=0\\ 17496 i \sqrt {3}\, b_{2}-17496 b_{2}&=0\\ -2592 i \sqrt {3}\, a_{2}+864 i \sqrt {3}\, b_{3}-2592 a_{2}+864 b_{3}&=0\\ -1296 i \sqrt {3}\, a_{2}+432 i \sqrt {3}\, b_{3}-1296 a_{2}+432 b_{3}&=0\\ 11664 i \sqrt {3}\, a_{2}-3888 i \sqrt {3}\, b_{3}+11664 a_{2}-3888 b_{3}&=0 \end {align*}
Solving the above equations for the unknowns gives \begin {align*} a_{1}&=0\\ a_{2}&=\frac {b_{3}}{3}\\ a_{3}&=0\\ b_{1}&=0\\ b_{2}&=0\\ b_{3}&=b_{3} \end {align*}
Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives \begin{align*} \xi &= \frac {x}{3} \\ \eta &= y \\ \end{align*} The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.
The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}
The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Unable to determine \(R\). Terminating
Unable to determine ODE type.
Solving equation (3)
Writing the ode as \begin {align*} y^{\prime }&=-\frac {2 x \left (6 i \sqrt {3}\, x^{2}-\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+6 x^{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}\\ y^{\prime }&= \omega \left ( x,y\right ) \end {align*}
The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end {align*}
The type of this ode is not in the lookup table. To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives \begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*} Where the unknown coefficients are \[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \] Substituting equations (1E,2E) and \(\omega \) into (A) gives \begin{equation} \tag{5E} b_{2}-\frac {2 x \left (6 i \sqrt {3}\, x^{2}-\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+6 x^{2}\right ) \left (b_{3}-a_{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}-\frac {4 x^{2} \left (6 i \sqrt {3}\, x^{2}-\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+6 x^{2}\right )^{2} a_{3}}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}-\left (-\frac {2 \left (6 i \sqrt {3}\, x^{2}-\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+6 x^{2}\right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}-\frac {2 x \left (12 i \sqrt {3}\, x +\frac {288 x^{5}}{\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \sqrt {-12 x^{6}+81 y^{2}}}+12 x \right )}{3 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}-\frac {96 x^{6} \left (6 i \sqrt {3}\, x^{2}-\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+6 x^{2}\right )}{\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \left (i \sqrt {3}-1\right ) \sqrt {-12 x^{6}+81 y^{2}}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\left (\frac {4 x \left (-108+\frac {972 y}{\sqrt {-12 x^{6}+81 y^{2}}}\right )}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}+\frac {2 x \left (6 i \sqrt {3}\, x^{2}-\left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}+6 x^{2}\right ) \left (-108+\frac {972 y}{\sqrt {-12 x^{6}+81 y^{2}}}\right )}{9 \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {4}{3}} \left (i \sqrt {3}-1\right )}\right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation} Putting the above in normal form gives \[ \text {Expression too large to display} \] Setting the numerator to zero gives \begin{equation} \tag{6E} \text {Expression too large to display} \end{equation} Simplifying the above gives \begin{equation} \tag{6E} \text {Expression too large to display} \end{equation} Since the PDE has radicals, simplifying gives \[ \text {Expression too large to display} \] Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them. \[ \left \{x, y, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}}, \sqrt {-12 x^{6}+81 y^{2}}\right \} \] The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them \[ \left \{x = v_{1}, y = v_{2}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}} = v_{3}, \left (-108 y +12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {2}{3}} = v_{4}, \sqrt {-12 x^{6}+81 y^{2}} = v_{5}\right \} \] The above PDE (6E) now becomes \begin{equation} \tag{7E} -432 i \sqrt {3}\, v_{4} v_{5} v_{1} v_{2} b_{3}-5184 i \sqrt {3}\, v_{3} v_{5} v_{1}^{4} a_{3} v_{2}+1296 i \sqrt {3}\, v_{4} v_{5} v_{1} a_{2} v_{2}+62208 v_{1}^{9} a_{2}+41472 v_{1}^{8} a_{1}-20736 v_{1}^{9} b_{3}+62208 v_{5} v_{1}^{3} a_{2} v_{2}+1728 i \sqrt {3}\, v_{4} v_{1}^{6} v_{2} a_{3}+46656 i \sqrt {3}\, v_{1}^{4} v_{3} v_{2}^{2} a_{3}-11664 i \sqrt {3}\, v_{1} v_{4} v_{2}^{2} a_{2}+3888 i \sqrt {3}\, v_{4} v_{1} v_{2}^{2} b_{3}+216 i \sqrt {3}\, v_{4} v_{5} v_{1}^{2} b_{2}-1944 i \sqrt {3}\, v_{4} v_{1}^{2} v_{2} b_{2}+216 i \sqrt {3}\, v_{4} v_{5} v_{1} b_{1}-1944 i \sqrt {3}\, v_{4} v_{1} v_{2} b_{1}+648 i \sqrt {3}\, v_{4} v_{5} v_{2}^{2} a_{3}+648 i \sqrt {3}\, v_{4} v_{5} a_{1} v_{2}+1944 i \sqrt {3}\, v_{3} v_{5} b_{2} v_{2}-288 i \sqrt {3}\, v_{4} v_{5} v_{1}^{6} a_{3}+46656 v_{5} v_{1}^{2} a_{1} v_{2}+288 v_{4} v_{5} v_{1}^{6} a_{3}-20736 v_{5} v_{1}^{3} v_{2} b_{3}-1728 v_{4} v_{1}^{6} v_{2} a_{3}-216 v_{4} v_{5} v_{1}^{2} b_{2}+1944 v_{4} v_{1}^{2} v_{2} b_{2}-3888 v_{4} v_{1} v_{2}^{2} b_{3}-216 v_{4} v_{5} v_{1} b_{1}+1944 v_{4} v_{1} v_{2} b_{1}+46656 v_{1}^{4} v_{3} v_{2}^{2} a_{3}+11664 v_{1} v_{4} v_{2}^{2} a_{2}-648 v_{4} v_{5} v_{2}^{2} a_{3}+1944 v_{3} v_{5} b_{2} v_{2}-648 v_{4} v_{5} a_{1} v_{2}-46656 v_{1}^{2} v_{5} v_{2}^{2} a_{3}+6912 v_{1}^{8} v_{5} a_{3}-6912 v_{1}^{10} v_{3} a_{3}+864 v_{1}^{7} v_{4} b_{3}+2592 v_{1}^{6} v_{3} b_{2}+5832 v_{4} v_{2}^{3} a_{3}+5832 v_{4} v_{2}^{2} a_{1}-17496 v_{3} v_{2}^{2} b_{2}-559872 v_{1}^{3} v_{2}^{2} a_{2}+419904 v_{1}^{2} v_{2}^{3} a_{3}-419904 v_{1}^{2} v_{2}^{2} a_{1}-82944 v_{1}^{8} v_{2} a_{3}+46656 v_{1}^{4} v_{2} b_{2}+186624 v_{1}^{3} v_{2}^{2} b_{3}+46656 v_{1}^{3} v_{2} b_{1}-5184 v_{5} v_{1}^{4} b_{2}-2592 v_{4} v_{1}^{7} a_{2}-5184 v_{5} v_{1}^{3} b_{1}-1728 v_{4} v_{1}^{6} a_{1}-6912 i \sqrt {3}\, v_{1}^{10} v_{3} a_{3}+2592 i \sqrt {3}\, v_{4} v_{1}^{7} a_{2}-864 i \sqrt {3}\, v_{1}^{7} v_{4} b_{3}+1728 i \sqrt {3}\, v_{4} v_{1}^{6} a_{1}+2592 i \sqrt {3}\, v_{1}^{6} v_{3} b_{2}-5832 i \sqrt {3}\, v_{4} v_{2}^{3} a_{3}-5832 i \sqrt {3}\, v_{4} v_{2}^{2} a_{1}-17496 i \sqrt {3}\, v_{3} v_{2}^{2} b_{2}-5184 v_{3} v_{5} v_{1}^{4} a_{3} v_{2}-1296 v_{4} v_{5} v_{1} a_{2} v_{2}+432 v_{4} v_{5} v_{1} v_{2} b_{3} = 0 \end{equation} Collecting the above on the terms \(v_i\) introduced, and these are \[ \{v_{1}, v_{2}, v_{3}, v_{4}, v_{5}\} \] Equation (7E) now becomes \begin{equation} \tag{8E} \left (62208 a_{2}-20736 b_{3}\right ) v_{1}^{9}+\left (-5184 i \sqrt {3}\, a_{3}-5184 a_{3}\right ) v_{1}^{4} v_{2} v_{3} v_{5}+\left (1296 i \sqrt {3}\, a_{2}-432 i \sqrt {3}\, b_{3}-1296 a_{2}+432 b_{3}\right ) v_{1} v_{2} v_{4} v_{5}+\left (-6912 i \sqrt {3}\, a_{3}-6912 a_{3}\right ) v_{1}^{10} v_{3}+\left (2592 i \sqrt {3}\, a_{2}-864 i \sqrt {3}\, b_{3}-2592 a_{2}+864 b_{3}\right ) v_{1}^{7} v_{4}+\left (2592 i \sqrt {3}\, b_{2}+2592 b_{2}\right ) v_{1}^{6} v_{3}+\left (1728 i \sqrt {3}\, a_{1}-1728 a_{1}\right ) v_{1}^{6} v_{4}+\left (-5832 i \sqrt {3}\, a_{3}+5832 a_{3}\right ) v_{2}^{3} v_{4}+\left (-17496 i \sqrt {3}\, b_{2}-17496 b_{2}\right ) v_{2}^{2} v_{3}+\left (-5832 i \sqrt {3}\, a_{1}+5832 a_{1}\right ) v_{2}^{2} v_{4}+\left (62208 a_{2}-20736 b_{3}\right ) v_{1}^{3} v_{2} v_{5}+41472 v_{1}^{8} a_{1}+\left (-559872 a_{2}+186624 b_{3}\right ) v_{1}^{3} v_{2}^{2}+46656 v_{5} v_{1}^{2} a_{1} v_{2}-46656 v_{1}^{2} v_{5} v_{2}^{2} a_{3}+6912 v_{1}^{8} v_{5} a_{3}+419904 v_{1}^{2} v_{2}^{3} a_{3}-419904 v_{1}^{2} v_{2}^{2} a_{1}-82944 v_{1}^{8} v_{2} a_{3}+46656 v_{1}^{4} v_{2} b_{2}+46656 v_{1}^{3} v_{2} b_{1}-5184 v_{5} v_{1}^{4} b_{2}-5184 v_{5} v_{1}^{3} b_{1}+\left (216 i \sqrt {3}\, b_{2}-216 b_{2}\right ) v_{1}^{2} v_{4} v_{5}+\left (-11664 i \sqrt {3}\, a_{2}+3888 i \sqrt {3}\, b_{3}+11664 a_{2}-3888 b_{3}\right ) v_{1} v_{2}^{2} v_{4}+\left (-1944 i \sqrt {3}\, b_{1}+1944 b_{1}\right ) v_{1} v_{2} v_{4}+\left (216 i \sqrt {3}\, b_{1}-216 b_{1}\right ) v_{1} v_{4} v_{5}+\left (648 i \sqrt {3}\, a_{3}-648 a_{3}\right ) v_{2}^{2} v_{4} v_{5}+\left (1944 i \sqrt {3}\, b_{2}+1944 b_{2}\right ) v_{2} v_{3} v_{5}+\left (648 i \sqrt {3}\, a_{1}-648 a_{1}\right ) v_{2} v_{4} v_{5}+\left (1728 i \sqrt {3}\, a_{3}-1728 a_{3}\right ) v_{1}^{6} v_{2} v_{4}+\left (-288 i \sqrt {3}\, a_{3}+288 a_{3}\right ) v_{1}^{6} v_{4} v_{5}+\left (46656 i \sqrt {3}\, a_{3}+46656 a_{3}\right ) v_{1}^{4} v_{2}^{2} v_{3}+\left (-1944 i \sqrt {3}\, b_{2}+1944 b_{2}\right ) v_{1}^{2} v_{2} v_{4} = 0 \end{equation} Setting each coefficients in (8E) to zero gives the following equations to solve \begin {align*} -419904 a_{1}&=0\\ 41472 a_{1}&=0\\ 46656 a_{1}&=0\\ -82944 a_{3}&=0\\ -46656 a_{3}&=0\\ 6912 a_{3}&=0\\ 419904 a_{3}&=0\\ -5184 b_{1}&=0\\ 46656 b_{1}&=0\\ -5184 b_{2}&=0\\ 46656 b_{2}&=0\\ -559872 a_{2}+186624 b_{3}&=0\\ 62208 a_{2}-20736 b_{3}&=0\\ -17496 i \sqrt {3}\, b_{2}-17496 b_{2}&=0\\ -6912 i \sqrt {3}\, a_{3}-6912 a_{3}&=0\\ -5832 i \sqrt {3}\, a_{1}+5832 a_{1}&=0\\ -5832 i \sqrt {3}\, a_{3}+5832 a_{3}&=0\\ -5184 i \sqrt {3}\, a_{3}-5184 a_{3}&=0\\ -1944 i \sqrt {3}\, b_{1}+1944 b_{1}&=0\\ -1944 i \sqrt {3}\, b_{2}+1944 b_{2}&=0\\ -288 i \sqrt {3}\, a_{3}+288 a_{3}&=0\\ 216 i \sqrt {3}\, b_{1}-216 b_{1}&=0\\ 216 i \sqrt {3}\, b_{2}-216 b_{2}&=0\\ 648 i \sqrt {3}\, a_{1}-648 a_{1}&=0\\ 648 i \sqrt {3}\, a_{3}-648 a_{3}&=0\\ 1728 i \sqrt {3}\, a_{1}-1728 a_{1}&=0\\ 1728 i \sqrt {3}\, a_{3}-1728 a_{3}&=0\\ 1944 i \sqrt {3}\, b_{2}+1944 b_{2}&=0\\ 2592 i \sqrt {3}\, b_{2}+2592 b_{2}&=0\\ 46656 i \sqrt {3}\, a_{3}+46656 a_{3}&=0\\ -11664 i \sqrt {3}\, a_{2}+3888 i \sqrt {3}\, b_{3}+11664 a_{2}-3888 b_{3}&=0\\ 1296 i \sqrt {3}\, a_{2}-432 i \sqrt {3}\, b_{3}-1296 a_{2}+432 b_{3}&=0\\ 2592 i \sqrt {3}\, a_{2}-864 i \sqrt {3}\, b_{3}-2592 a_{2}+864 b_{3}&=0 \end {align*}
Solving the above equations for the unknowns gives \begin {align*} a_{1}&=0\\ a_{2}&=\frac {b_{3}}{3}\\ a_{3}&=0\\ b_{1}&=0\\ b_{2}&=0\\ b_{3}&=b_{3} \end {align*}
Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives \begin{align*} \xi &= \frac {x}{3} \\ \eta &= y \\ \end{align*} The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.
The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}
The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Unable to determine \(R\). Terminating
Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}-4 y^{\prime } x^{4}+8 y x^{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 }=2 \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right ) x , y^{\prime }=2 \left (-\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x , y^{\prime }=2 \left (-\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=2 \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right ) x \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=2 \left (-\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=2 \left (-\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (-108 y+12 \sqrt {-12 x^{6}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: *** Sublevel 2 *** 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 dAlembert trying simple symmetries for implicit equations Successful isolation of dy/dx: 3 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** 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 -> Calling odsolve with the ODE`, diff(y(x), x) = (-4*y(x)^5+3*y(x)*x^2)/(-4*y(x)^4*x+3*x^3), y(x)` *** Sublevel 3 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- 1st order, parametric methods successful`
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 40
dsolve(diff(y(x),x)^3-4*x^4*diff(y(x),x)+8*x^3*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {2 \sqrt {3}\, x^{3}}{9} \\ y \left (x \right ) &= \frac {2 \sqrt {3}\, x^{3}}{9} \\ y \left (x \right ) &= \frac {x^{2}}{2 c_{1}}-\frac {1}{8 c_{1}^{3}} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]^3-4*x^4*y'[x]+8*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Timed out