Internal problem ID [9272]
Internal file name [OUTPUT/8208_Monday_June_06_2022_02_17_30_AM_3103180/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 939.
ODE order: 1.
ODE degree: 1.
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], _rational, [_Abel, `2nd type`, `class C`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {-32 y x +16 x^{3}+16 x^{2}-32 x -64 y^{3}+48 y^{2} x^{2}+96 y^{2} x -12 x^{4} y-48 y x^{3}-48 y x^{2}+x^{6}+6 x^{5}+12 x^{4}}{-64 y+16 x^{2}+32 x -64}=0} \]
Writing the ode as \begin {align*} y^{\prime }&=\frac {-x^{6}-6 x^{5}+12 x^{4} y -12 x^{4}+48 y \,x^{3}-48 y^{2} x^{2}-16 x^{3}+48 x^{2} y -96 y^{2} x +64 y^{3}-16 x^{2}+32 y x +32 x}{-16 x^{2}-32 x +64 y +64}\\ 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 {\left (-x^{6}-6 x^{5}+12 x^{4} y -12 x^{4}+48 y \,x^{3}-48 y^{2} x^{2}-16 x^{3}+48 x^{2} y -96 y^{2} x +64 y^{3}-16 x^{2}+32 y x +32 x \right ) \left (b_{3}-a_{2}\right )}{-16 x^{2}-32 x +64 y +64}-\frac {\left (-x^{6}-6 x^{5}+12 x^{4} y -12 x^{4}+48 y \,x^{3}-48 y^{2} x^{2}-16 x^{3}+48 x^{2} y -96 y^{2} x +64 y^{3}-16 x^{2}+32 y x +32 x \right )^{2} a_{3}}{256 \left (-x^{2}-2 x +4 y +4\right )^{2}}-\left (\frac {-6 x^{5}-30 x^{4}+48 y \,x^{3}-48 x^{3}+144 x^{2} y -96 y^{2} x -48 x^{2}+96 y x -96 y^{2}-32 x +32 y +32}{-16 x^{2}-32 x +64 y +64}-\frac {\left (-x^{6}-6 x^{5}+12 x^{4} y -12 x^{4}+48 y \,x^{3}-48 y^{2} x^{2}-16 x^{3}+48 x^{2} y -96 y^{2} x +64 y^{3}-16 x^{2}+32 y x +32 x \right ) \left (-2 x -2\right )}{16 \left (-x^{2}-2 x +4 y +4\right )^{2}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\left (\frac {12 x^{4}+48 x^{3}-96 x^{2} y +48 x^{2}-192 y x +192 y^{2}+32 x}{-16 x^{2}-32 x +64 y +64}-\frac {-x^{6}-6 x^{5}+12 x^{4} y -12 x^{4}+48 y \,x^{3}-48 y^{2} x^{2}-16 x^{3}+48 x^{2} y -96 y^{2} x +64 y^{3}-16 x^{2}+32 y x +32 x}{4 \left (-x^{2}-2 x +4 y +4\right )^{2}}\right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation} Putting the above in normal form gives \[ -\frac {x^{12} a_{3}+12 x^{11} a_{3}-24 x^{10} y a_{3}+60 x^{10} a_{3}-240 x^{9} y a_{3}+240 x^{8} y^{2} a_{3}+176 x^{9} a_{3}-960 x^{8} y a_{3}+1920 x^{7} y^{2} a_{3}-1280 x^{6} y^{3} a_{3}+80 x^{8} a_{2}+368 x^{8} a_{3}-16 x^{8} b_{3}-2112 x^{7} y a_{3}+5760 x^{6} y^{2} a_{3}-7680 x^{5} y^{3} a_{3}+3840 x^{4} y^{4} a_{3}+64 x^{7} a_{1}+576 x^{7} a_{2}+512 x^{7} a_{3}-128 x^{7} b_{2}-128 x^{7} b_{3}-1024 x^{6} y a_{2}-3008 x^{6} y a_{3}+128 x^{6} y b_{3}+8448 x^{5} y^{2} a_{3}-15360 x^{4} y^{3} a_{3}+15360 x^{3} y^{4} a_{3}-6144 x^{2} y^{5} a_{3}+448 x^{6} a_{1}+1088 x^{6} a_{2}+256 x^{6} a_{3}-128 x^{6} b_{1}-768 x^{6} b_{2}-320 x^{6} b_{3}-768 x^{5} y a_{1}-5376 x^{5} y a_{2}-2304 x^{5} y a_{3}+1536 x^{5} y b_{2}+768 x^{5} y b_{3}+4608 x^{4} y^{2} a_{2}+6144 x^{4} y^{2} a_{3}-11264 x^{3} y^{3} a_{3}+15360 x^{2} y^{4} a_{3}-12288 x \,y^{5} a_{3}+4096 y^{6} a_{3}+768 x^{5} a_{1}-256 x^{5} a_{2}-256 x^{5} a_{3}-768 x^{5} b_{1}-768 x^{5} b_{2}-256 x^{5} b_{3}-3840 x^{4} y a_{1}-5376 x^{4} y a_{2}+1536 x^{4} y b_{1}+6144 x^{4} y b_{2}+1536 x^{4} y b_{3}+3072 x^{3} y^{2} a_{1}+15360 x^{3} y^{2} a_{2}-6144 x^{3} y^{2} b_{2}-8192 x^{2} y^{3} a_{2}+1024 x^{2} y^{3} a_{3}-2048 x^{2} y^{3} b_{3}-512 x^{4} a_{1}-2048 x^{4} a_{2}-768 x^{4} a_{3}-768 x^{4} b_{1}+1792 x^{4} b_{2}-3072 x^{3} y a_{1}+5120 x^{3} y a_{2}+6144 x^{3} y b_{1}+2048 x^{3} y b_{3}+9216 x^{2} y^{2} a_{1}+3072 x^{2} y^{2} a_{2}-6144 x^{2} y^{2} b_{1}-12288 x^{2} y^{2} b_{2}-3072 x^{2} y^{2} b_{3}-4096 x \,y^{3} a_{1}-12288 x \,y^{3} a_{2}+4096 x \,y^{3} a_{3}+8192 x \,y^{3} b_{2}-4096 x \,y^{3} b_{3}+4096 y^{4} a_{2}-4096 y^{4} a_{3}+4096 y^{4} b_{3}-2048 x^{3} a_{1}-3072 x^{3} a_{2}-1024 x^{3} a_{3}+2048 x^{3} b_{1}+2048 x^{3} b_{2}+1024 x^{3} b_{3}+5120 x^{2} y a_{1}+5120 x^{2} y a_{2}-10240 x^{2} y b_{2}+2048 x^{2} y b_{3}-8192 x \,y^{2} a_{2}+4096 x \,y^{2} a_{3}-12288 x \,y^{2} b_{1}+12288 x \,y^{2} b_{2}-8192 x \,y^{2} b_{3}-4096 y^{3} a_{1}+4096 y^{3} a_{2}-4096 y^{3} a_{3}+8192 y^{3} b_{1}+8192 y^{3} b_{3}-2048 x^{2} a_{1}-4096 x^{2} a_{2}+1024 x^{2} a_{3}+3072 x^{2} b_{1}+1024 x^{2} b_{2}+2048 x^{2} b_{3}+4096 x y a_{1}+8192 x y a_{2}-2048 x y a_{3}-12288 x y b_{1}+4096 x y b_{2}-4096 x y b_{3}-4096 y^{2} a_{1}+4096 y^{2} a_{3}+12288 y^{2} b_{1}-4096 y^{2} b_{2}-2048 x a_{1}+4096 x a_{2}+4096 x b_{2}-2048 x b_{3}+4096 y a_{1}+2048 y a_{3}-8192 y b_{2}+2048 a_{1}-4096 b_{2}}{256 \left (x^{2}+2 x -4 y -4\right )^{2}} = 0 \] Setting the numerator to zero gives \begin{equation} \tag{6E} -x^{12} a_{3}-12 x^{11} a_{3}+24 x^{10} y a_{3}-60 x^{10} a_{3}+240 x^{9} y a_{3}-240 x^{8} y^{2} a_{3}-176 x^{9} a_{3}+960 x^{8} y a_{3}-1920 x^{7} y^{2} a_{3}+1280 x^{6} y^{3} a_{3}-80 x^{8} a_{2}-368 x^{8} a_{3}+16 x^{8} b_{3}+2112 x^{7} y a_{3}-5760 x^{6} y^{2} a_{3}+7680 x^{5} y^{3} a_{3}-3840 x^{4} y^{4} a_{3}-64 x^{7} a_{1}-576 x^{7} a_{2}-512 x^{7} a_{3}+128 x^{7} b_{2}+128 x^{7} b_{3}+1024 x^{6} y a_{2}+3008 x^{6} y a_{3}-128 x^{6} y b_{3}-8448 x^{5} y^{2} a_{3}+15360 x^{4} y^{3} a_{3}-15360 x^{3} y^{4} a_{3}+6144 x^{2} y^{5} a_{3}-448 x^{6} a_{1}-1088 x^{6} a_{2}-256 x^{6} a_{3}+128 x^{6} b_{1}+768 x^{6} b_{2}+320 x^{6} b_{3}+768 x^{5} y a_{1}+5376 x^{5} y a_{2}+2304 x^{5} y a_{3}-1536 x^{5} y b_{2}-768 x^{5} y b_{3}-4608 x^{4} y^{2} a_{2}-6144 x^{4} y^{2} a_{3}+11264 x^{3} y^{3} a_{3}-15360 x^{2} y^{4} a_{3}+12288 x \,y^{5} a_{3}-4096 y^{6} a_{3}-768 x^{5} a_{1}+256 x^{5} a_{2}+256 x^{5} a_{3}+768 x^{5} b_{1}+768 x^{5} b_{2}+256 x^{5} b_{3}+3840 x^{4} y a_{1}+5376 x^{4} y a_{2}-1536 x^{4} y b_{1}-6144 x^{4} y b_{2}-1536 x^{4} y b_{3}-3072 x^{3} y^{2} a_{1}-15360 x^{3} y^{2} a_{2}+6144 x^{3} y^{2} b_{2}+8192 x^{2} y^{3} a_{2}-1024 x^{2} y^{3} a_{3}+2048 x^{2} y^{3} b_{3}+512 x^{4} a_{1}+2048 x^{4} a_{2}+768 x^{4} a_{3}+768 x^{4} b_{1}-1792 x^{4} b_{2}+3072 x^{3} y a_{1}-5120 x^{3} y a_{2}-6144 x^{3} y b_{1}-2048 x^{3} y b_{3}-9216 x^{2} y^{2} a_{1}-3072 x^{2} y^{2} a_{2}+6144 x^{2} y^{2} b_{1}+12288 x^{2} y^{2} b_{2}+3072 x^{2} y^{2} b_{3}+4096 x \,y^{3} a_{1}+12288 x \,y^{3} a_{2}-4096 x \,y^{3} a_{3}-8192 x \,y^{3} b_{2}+4096 x \,y^{3} b_{3}-4096 y^{4} a_{2}+4096 y^{4} a_{3}-4096 y^{4} b_{3}+2048 x^{3} a_{1}+3072 x^{3} a_{2}+1024 x^{3} a_{3}-2048 x^{3} b_{1}-2048 x^{3} b_{2}-1024 x^{3} b_{3}-5120 x^{2} y a_{1}-5120 x^{2} y a_{2}+10240 x^{2} y b_{2}-2048 x^{2} y b_{3}+8192 x \,y^{2} a_{2}-4096 x \,y^{2} a_{3}+12288 x \,y^{2} b_{1}-12288 x \,y^{2} b_{2}+8192 x \,y^{2} b_{3}+4096 y^{3} a_{1}-4096 y^{3} a_{2}+4096 y^{3} a_{3}-8192 y^{3} b_{1}-8192 y^{3} b_{3}+2048 x^{2} a_{1}+4096 x^{2} a_{2}-1024 x^{2} a_{3}-3072 x^{2} b_{1}-1024 x^{2} b_{2}-2048 x^{2} b_{3}-4096 x y a_{1}-8192 x y a_{2}+2048 x y a_{3}+12288 x y b_{1}-4096 x y b_{2}+4096 x y b_{3}+4096 y^{2} a_{1}-4096 y^{2} a_{3}-12288 y^{2} b_{1}+4096 y^{2} b_{2}+2048 x a_{1}-4096 x a_{2}-4096 x b_{2}+2048 x b_{3}-4096 y a_{1}-2048 y a_{3}+8192 y b_{2}-2048 a_{1}+4096 b_{2} = 0 \end{equation} Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them. \[ \{x, y\} \] The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them \[ \{x = v_{1}, y = v_{2}\} \] The above PDE (6E) now becomes \begin{equation} \tag{7E} \text {Expression too large to display} \end{equation} Collecting the above on the terms \(v_i\) introduced, and these are \[ \{v_{1}, v_{2}\} \] Equation (7E) now becomes \begin{equation} \tag{8E} \left (1024 a_{2}+3008 a_{3}-128 b_{3}\right ) v_{1}^{6} v_{2}+\left (768 a_{1}+5376 a_{2}+2304 a_{3}-1536 b_{2}-768 b_{3}\right ) v_{1}^{5} v_{2}+\left (-4608 a_{2}-6144 a_{3}\right ) v_{1}^{4} v_{2}^{2}+\left (3840 a_{1}+5376 a_{2}-1536 b_{1}-6144 b_{2}-1536 b_{3}\right ) v_{1}^{4} v_{2}+\left (-3072 a_{1}-15360 a_{2}+6144 b_{2}\right ) v_{1}^{3} v_{2}^{2}+\left (3072 a_{1}-5120 a_{2}-6144 b_{1}-2048 b_{3}\right ) v_{1}^{3} v_{2}+\left (8192 a_{2}-1024 a_{3}+2048 b_{3}\right ) v_{1}^{2} v_{2}^{3}+\left (-9216 a_{1}-3072 a_{2}+6144 b_{1}+12288 b_{2}+3072 b_{3}\right ) v_{1}^{2} v_{2}^{2}+\left (-5120 a_{1}-5120 a_{2}+10240 b_{2}-2048 b_{3}\right ) v_{1}^{2} v_{2}+\left (4096 a_{1}+12288 a_{2}-4096 a_{3}-8192 b_{2}+4096 b_{3}\right ) v_{1} v_{2}^{3}+\left (8192 a_{2}-4096 a_{3}+12288 b_{1}-12288 b_{2}+8192 b_{3}\right ) v_{1} v_{2}^{2}+\left (-4096 a_{1}-8192 a_{2}+2048 a_{3}+12288 b_{1}-4096 b_{2}+4096 b_{3}\right ) v_{1} v_{2}-2048 a_{1}+4096 b_{2}+\left (-80 a_{2}-368 a_{3}+16 b_{3}\right ) v_{1}^{8}+\left (-64 a_{1}-576 a_{2}-512 a_{3}+128 b_{2}+128 b_{3}\right ) v_{1}^{7}+\left (-448 a_{1}-1088 a_{2}-256 a_{3}+128 b_{1}+768 b_{2}+320 b_{3}\right ) v_{1}^{6}-a_{3} v_{1}^{12}-12 a_{3} v_{1}^{11}-60 a_{3} v_{1}^{10}-176 a_{3} v_{1}^{9}-4096 a_{3} v_{2}^{6}+\left (2048 a_{1}+4096 a_{2}-1024 a_{3}-3072 b_{1}-1024 b_{2}-2048 b_{3}\right ) v_{1}^{2}+\left (2048 a_{1}-4096 a_{2}-4096 b_{2}+2048 b_{3}\right ) v_{1}+\left (-4096 a_{2}+4096 a_{3}-4096 b_{3}\right ) v_{2}^{4}+\left (4096 a_{1}-4096 a_{2}+4096 a_{3}-8192 b_{1}-8192 b_{3}\right ) v_{2}^{3}+\left (4096 a_{1}-4096 a_{3}-12288 b_{1}+4096 b_{2}\right ) v_{2}^{2}+\left (-4096 a_{1}-2048 a_{3}+8192 b_{2}\right ) v_{2}+24 a_{3} v_{1}^{10} v_{2}+240 a_{3} v_{1}^{9} v_{2}-240 a_{3} v_{1}^{8} v_{2}^{2}+960 a_{3} v_{1}^{8} v_{2}-1920 a_{3} v_{1}^{7} v_{2}^{2}+1280 a_{3} v_{1}^{6} v_{2}^{3}+2112 a_{3} v_{1}^{7} v_{2}-5760 a_{3} v_{1}^{6} v_{2}^{2}+7680 a_{3} v_{1}^{5} v_{2}^{3}-3840 a_{3} v_{1}^{4} v_{2}^{4}-8448 a_{3} v_{1}^{5} v_{2}^{2}+15360 a_{3} v_{1}^{4} v_{2}^{3}-15360 a_{3} v_{1}^{3} v_{2}^{4}+6144 a_{3} v_{1}^{2} v_{2}^{5}+11264 a_{3} v_{1}^{3} v_{2}^{3}-15360 a_{3} v_{1}^{2} v_{2}^{4}+12288 a_{3} v_{1} v_{2}^{5}+\left (-768 a_{1}+256 a_{2}+256 a_{3}+768 b_{1}+768 b_{2}+256 b_{3}\right ) v_{1}^{5}+\left (512 a_{1}+2048 a_{2}+768 a_{3}+768 b_{1}-1792 b_{2}\right ) v_{1}^{4}+\left (2048 a_{1}+3072 a_{2}+1024 a_{3}-2048 b_{1}-2048 b_{2}-1024 b_{3}\right ) v_{1}^{3} = 0 \end{equation} Setting each coefficients in (8E) to zero gives the following equations to solve \begin {align*} -15360 a_{3}&=0\\ -8448 a_{3}&=0\\ -5760 a_{3}&=0\\ -4096 a_{3}&=0\\ -3840 a_{3}&=0\\ -1920 a_{3}&=0\\ -240 a_{3}&=0\\ -176 a_{3}&=0\\ -60 a_{3}&=0\\ -12 a_{3}&=0\\ -a_{3}&=0\\ 24 a_{3}&=0\\ 240 a_{3}&=0\\ 960 a_{3}&=0\\ 1280 a_{3}&=0\\ 2112 a_{3}&=0\\ 6144 a_{3}&=0\\ 7680 a_{3}&=0\\ 11264 a_{3}&=0\\ 12288 a_{3}&=0\\ 15360 a_{3}&=0\\ -2048 a_{1}+4096 b_{2}&=0\\ -4608 a_{2}-6144 a_{3}&=0\\ -4096 a_{1}-2048 a_{3}+8192 b_{2}&=0\\ -3072 a_{1}-15360 a_{2}+6144 b_{2}&=0\\ -4096 a_{2}+4096 a_{3}-4096 b_{3}&=0\\ -80 a_{2}-368 a_{3}+16 b_{3}&=0\\ 1024 a_{2}+3008 a_{3}-128 b_{3}&=0\\ 8192 a_{2}-1024 a_{3}+2048 b_{3}&=0\\ -5120 a_{1}-5120 a_{2}+10240 b_{2}-2048 b_{3}&=0\\ 2048 a_{1}-4096 a_{2}-4096 b_{2}+2048 b_{3}&=0\\ 3072 a_{1}-5120 a_{2}-6144 b_{1}-2048 b_{3}&=0\\ 4096 a_{1}-4096 a_{3}-12288 b_{1}+4096 b_{2}&=0\\ -9216 a_{1}-3072 a_{2}+6144 b_{1}+12288 b_{2}+3072 b_{3}&=0\\ -64 a_{1}-576 a_{2}-512 a_{3}+128 b_{2}+128 b_{3}&=0\\ 512 a_{1}+2048 a_{2}+768 a_{3}+768 b_{1}-1792 b_{2}&=0\\ 768 a_{1}+5376 a_{2}+2304 a_{3}-1536 b_{2}-768 b_{3}&=0\\ 3840 a_{1}+5376 a_{2}-1536 b_{1}-6144 b_{2}-1536 b_{3}&=0\\ 4096 a_{1}-4096 a_{2}+4096 a_{3}-8192 b_{1}-8192 b_{3}&=0\\ 4096 a_{1}+12288 a_{2}-4096 a_{3}-8192 b_{2}+4096 b_{3}&=0\\ 8192 a_{2}-4096 a_{3}+12288 b_{1}-12288 b_{2}+8192 b_{3}&=0\\ -4096 a_{1}-8192 a_{2}+2048 a_{3}+12288 b_{1}-4096 b_{2}+4096 b_{3}&=0\\ -768 a_{1}+256 a_{2}+256 a_{3}+768 b_{1}+768 b_{2}+256 b_{3}&=0\\ -448 a_{1}-1088 a_{2}-256 a_{3}+128 b_{1}+768 b_{2}+320 b_{3}&=0\\ 2048 a_{1}+3072 a_{2}+1024 a_{3}-2048 b_{1}-2048 b_{2}-1024 b_{3}&=0\\ 2048 a_{1}+4096 a_{2}-1024 a_{3}-3072 b_{1}-1024 b_{2}-2048 b_{3}&=0 \end {align*}
Solving the above equations for the unknowns gives \begin {align*} a_{1}&=a_{1}\\ a_{2}&=0\\ a_{3}&=0\\ b_{1}&=\frac {a_{1}}{2}\\ b_{2}&=\frac {a_{1}}{2}\\ b_{3}&=0 \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 &= 1 \\ \eta &= \frac {x}{2}+\frac {1}{2} \\ \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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -x^{6}+12 x^{4} y-6 x^{5}-48 y^{2} x^{2}+48 y x^{3}-12 x^{4}+64 y^{3}-96 y^{2} x +48 y x^{2}+16 x^{2} y^{\prime }-16 x^{3}-64 y y^{\prime }+32 y x +32 y^{\prime } x -16 x^{2}-64 y^{\prime }+32 x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-32 y x +16 x^{3}+16 x^{2}-32 x -64 y^{3}+48 y^{2} x^{2}+96 y^{2} x -12 x^{4} y-48 y x^{3}-48 y x^{2}+x^{6}+6 x^{5}+12 x^{4}}{-64 y+16 x^{2}+32 x -64} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries differential order: 1; found: 1 linear symmetries. Trying reduction of order 1st order, trying the canonical coordinates of the invariance group <- 1st order, canonical coordinates successful`
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 79
dsolve(diff(y(x),x) = (-32*x*y(x)+16*x^3+16*x^2-32*x-64*y(x)^3+48*x^2*y(x)^2+96*x*y(x)^2-12*y(x)*x^4-48*x^3*y(x)-48*x^2*y(x)+x^6+6*x^5+12*x^4)/(-64*y(x)+16*x^2+32*x-64),y(x), singsol=all)
\[ x +\frac {2 \ln \left (2\right )}{5}+\frac {2 \ln \left (16 y \left (x \right )^{2}+\left (-8 x^{2}-16 x +16\right ) y \left (x \right )+x^{4}+4 x^{3}-8 x +8\right )}{5}-\frac {2 \arctan \left (-2 y \left (x \right )+\frac {x^{2}}{2}+x -1\right )}{5}-\frac {4 \ln \left (4 y \left (x \right )-x^{2}-2 x -4\right )}{5}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.374 (sec). Leaf size: 136
DSolve[y'[x] == (-32*x + 16*x^2 + 16*x^3 + 12*x^4 + 6*x^5 + x^6 - 32*x*y[x] - 48*x^2*y[x] - 48*x^3*y[x] - 12*x^4*y[x] + 96*x*y[x]^2 + 48*x^2*y[x]^2 - 64*y[x]^3)/(-64 + 32*x + 16*x^2 - 64*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2}{5} \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3-8 \text {$\#$1}^2 y(x)-16 \text {$\#$1} y(x)-8 \text {$\#$1}+16 y(x)^2+16 y(x)+8\&,\frac {\text {$\#$1}^2 (-\log (x-\text {$\#$1}))+4 y(x) \log (x-\text {$\#$1})-2 \text {$\#$1} \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1})}{-\text {$\#$1}^2-2 \text {$\#$1}+4 y(x)+2}\&\right ]-\frac {4}{5} \log \left (x^2-4 y(x)+2 x+4\right )+x=c_1,y(x)\right ] \]