Internal problem ID [9055]
Internal file name [OUTPUT/7990_Monday_June_06_2022_01_08_46_AM_88785126/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 721.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati", "first_order_ode_lie_symmetry_calculated"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {y^{\prime }-\frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y x^{3}+x^{6}\right ) \sqrt {x}}{36}=0} \]
Writing the ode as \begin {align*} y^{\prime }&=\frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y \,x^{3}+x^{6}\right ) \sqrt {x}}{36}\\ 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 3 to use as anstaz gives \begin{align*} \tag{1E} \xi &= x^{3} a_{7}+x^{2} y a_{8}+y^{2} x a_{9}+y^{3} a_{10}+x^{2} a_{4}+y x a_{5}+y^{2} a_{6}+x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x^{3} b_{7}+x^{2} y b_{8}+y^{2} x b_{9}+y^{3} b_{10}+x^{2} b_{4}+y x b_{5}+y^{2} b_{6}+x b_{2}+y b_{3}+b_{1} \\ \end{align*} Where the unknown coefficients are \[ \{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}, a_{9}, a_{10}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}, b_{8}, b_{9}, b_{10}\} \] Substituting equations (1E,2E) and \(\omega \) into (A) gives \begin{equation} \tag{5E} 3 x^{2} b_{7}+2 x y b_{8}+y^{2} b_{9}+2 x b_{4}+y b_{5}+b_{2}+\frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y \,x^{3}+x^{6}\right ) \sqrt {x}\, \left (-3 x^{2} a_{7}+x^{2} b_{8}-2 x y a_{8}+2 x y b_{9}-y^{2} a_{9}+3 y^{2} b_{10}-2 x a_{4}+x b_{5}-y a_{5}+2 y b_{6}-a_{2}+b_{3}\right )}{36}-\frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y \,x^{3}+x^{6}\right )^{2} x \left (x^{2} a_{8}+2 x y a_{9}+3 y^{2} a_{10}+x a_{5}+2 y a_{6}+a_{3}\right )}{1296}-\left (\frac {\left (27 \sqrt {x}-36 x^{2} y +6 x^{5}\right ) \sqrt {x}}{36}+\frac {18 x^{{3}/{2}}+36 y^{2}-12 y \,x^{3}+x^{6}}{72 \sqrt {x}}\right ) \left (x^{3} a_{7}+x^{2} y a_{8}+y^{2} x a_{9}+y^{3} a_{10}+x^{2} a_{4}+y x a_{5}+y^{2} a_{6}+x a_{2}+y a_{3}+a_{1}\right )-\frac {\left (-12 x^{3}+72 y \right ) \sqrt {x}\, \left (x^{3} b_{7}+x^{2} y b_{8}+y^{2} x b_{9}+y^{3} b_{10}+x^{2} b_{4}+y x b_{5}+y^{2} b_{6}+x b_{2}+y b_{3}+b_{1}\right )}{36} = 0 \end{equation} Putting the above in normal form gives \[ -\frac {-108 x^{7} y^{2} b_{10}+432 x^{5} y^{2} b_{9}+864 x^{4} y^{3} b_{10}+2592 x^{{7}/{2}} y a_{8}-1296 x^{{7}/{2}} y b_{9}+1296 x^{3} y^{2} b_{8}+2 x^{{29}/{2}} y a_{9}+3 x^{{27}/{2}} y^{2} a_{10}-24 x^{{25}/{2}} y a_{8}-48 x^{{23}/{2}} y^{2} a_{9}-72 x^{{21}/{2}} y^{3} a_{10}+72 x^{10} y a_{9}+108 x^{9} y^{2} a_{10}+216 x^{{19}/{2}} y^{2} a_{8}+432 x^{{17}/{2}} y^{3} a_{9}+648 x^{{15}/{2}} y^{4} a_{10}+648 x^{{11}/{2}} y a_{9}+972 x^{{9}/{2}} y^{2} a_{10}-864 x^{{13}/{2}} y^{3} a_{8}-1728 x^{{11}/{2}} y^{4} a_{9}-2592 x^{{9}/{2}} y^{5} a_{10}+1296 x^{{7}/{2}} y^{4} a_{8}+2592 x^{{5}/{2}} y^{5} a_{9}+3888 x^{{3}/{2}} y^{6} a_{10}+1944 x^{{5}/{2}} y^{2} a_{9}+2592 x^{4} y b_{7}-2592 x^{{3}/{2}} y b_{8}-72 x^{8} y b_{9}-24 x^{{21}/{2}} y a_{3}+216 x^{{15}/{2}} y^{2} a_{3}-864 x^{{9}/{2}} y^{3} a_{3}-72 x^{7} y b_{6}+432 x^{4} y^{2} b_{6}+1296 x^{2} y^{2} b_{5}+2 x^{{27}/{2}} y a_{6}-24 x^{{23}/{2}} y a_{5}-48 x^{{21}/{2}} y^{2} a_{6}+72 x^{9} y a_{6}+216 x^{{17}/{2}} y^{2} a_{5}+432 x^{{15}/{2}} y^{3} a_{6}+648 x^{{9}/{2}} y a_{6}-864 x^{{11}/{2}} y^{3} a_{5}-1728 x^{{9}/{2}} y^{4} a_{6}+1296 x^{{5}/{2}} y^{4} a_{5}+2592 x^{{3}/{2}} y^{5} a_{6}+1944 x^{{5}/{2}} y a_{5}+2592 x^{3} y b_{4}+1296 x^{{3}/{2}} y^{4} a_{3}+2592 x^{2} y b_{2}-198 x^{6} y a_{3}-1944 x^{4} y a_{2}-216 x^{3} y^{2} a_{3}-1512 x^{3} y a_{1}+1944 x \,y^{2} a_{2}+x^{{27}/{2}} a_{3}+36 x^{9} a_{3}+324 x^{{9}/{2}} a_{3}-36 x^{7} b_{3}-432 x^{5} b_{2}-432 x^{4} b_{1}+1296 x^{{3}/{2}} y a_{3}+1296 x \,y^{2} b_{3}+2592 x y b_{1}+342 x^{9} a_{7}+648 y^{5} a_{10}+270 x^{7} a_{2}+234 x^{6} a_{1}+648 y^{3} a_{3}+648 y^{2} a_{1}+1944 x^{{5}/{2}} a_{2}-648 x^{{5}/{2}} b_{3}+1296 x^{{3}/{2}} a_{1}-1296 b_{2} \sqrt {x}+306 x^{8} a_{4}+648 y^{4} a_{6}-162 x^{7} y a_{5}-630 x^{6} y^{2} a_{6}-2376 x^{5} y a_{4}-648 x^{4} y^{2} a_{5}+1080 x^{3} y^{3} a_{6}+3240 x^{2} y^{2} a_{4}+1944 x \,y^{3} a_{5}-2592 x^{{3}/{2}} b_{4}-1296 y b_{5} \sqrt {x}-36 x^{8} b_{5}+2592 x^{{7}/{2}} a_{4}-648 x^{{7}/{2}} b_{5}-1296 x^{{5}/{2}} y b_{6}+x^{{29}/{2}} a_{5}+36 x^{10} a_{5}+324 x^{{11}/{2}} a_{5}+1296 x^{{3}/{2}} y^{2} a_{6}-432 x^{6} b_{4}-1944 x^{{5}/{2}} y^{2} b_{10}-1296 x \,y^{4} b_{10}+x^{{31}/{2}} a_{8}+36 x^{11} a_{8}+324 x^{{13}/{2}} a_{8}+3240 x^{{9}/{2}} a_{7}+1296 x^{{3}/{2}} y^{3} a_{10}-432 x^{7} b_{7}-3888 x^{{5}/{2}} b_{7}-1296 y^{2} b_{9} \sqrt {x}-36 x^{9} b_{8}-648 x^{{9}/{2}} b_{8}-126 x^{8} y a_{8}-594 x^{7} y^{2} a_{9}-1062 x^{6} y^{3} a_{10}-2808 x^{6} y a_{7}-1080 x^{5} y^{2} a_{8}+648 x^{4} y^{3} a_{9}+2376 x^{3} y^{4} a_{10}+4536 x^{3} y^{2} a_{7}+3240 x^{2} y^{3} a_{8}+1944 x \,y^{4} a_{9}}{1296 \sqrt {x}} = 0 \] Setting the numerator to zero gives \begin{equation} \tag{6E} 108 x^{7} y^{2} b_{10}-432 x^{5} y^{2} b_{9}-864 x^{4} y^{3} b_{10}-2592 x^{{7}/{2}} y a_{8}+1296 x^{{7}/{2}} y b_{9}-1296 x^{3} y^{2} b_{8}-2 x^{{29}/{2}} y a_{9}-3 x^{{27}/{2}} y^{2} a_{10}+24 x^{{25}/{2}} y a_{8}+48 x^{{23}/{2}} y^{2} a_{9}+72 x^{{21}/{2}} y^{3} a_{10}-72 x^{10} y a_{9}-108 x^{9} y^{2} a_{10}-216 x^{{19}/{2}} y^{2} a_{8}-432 x^{{17}/{2}} y^{3} a_{9}-648 x^{{15}/{2}} y^{4} a_{10}-648 x^{{11}/{2}} y a_{9}-972 x^{{9}/{2}} y^{2} a_{10}+864 x^{{13}/{2}} y^{3} a_{8}+1728 x^{{11}/{2}} y^{4} a_{9}+2592 x^{{9}/{2}} y^{5} a_{10}-1296 x^{{7}/{2}} y^{4} a_{8}-2592 x^{{5}/{2}} y^{5} a_{9}-3888 x^{{3}/{2}} y^{6} a_{10}-1944 x^{{5}/{2}} y^{2} a_{9}-2592 x^{4} y b_{7}+2592 x^{{3}/{2}} y b_{8}+72 x^{8} y b_{9}+24 x^{{21}/{2}} y a_{3}-216 x^{{15}/{2}} y^{2} a_{3}+864 x^{{9}/{2}} y^{3} a_{3}+72 x^{7} y b_{6}-432 x^{4} y^{2} b_{6}-1296 x^{2} y^{2} b_{5}-2 x^{{27}/{2}} y a_{6}+24 x^{{23}/{2}} y a_{5}+48 x^{{21}/{2}} y^{2} a_{6}-72 x^{9} y a_{6}-216 x^{{17}/{2}} y^{2} a_{5}-432 x^{{15}/{2}} y^{3} a_{6}-648 x^{{9}/{2}} y a_{6}+864 x^{{11}/{2}} y^{3} a_{5}+1728 x^{{9}/{2}} y^{4} a_{6}-1296 x^{{5}/{2}} y^{4} a_{5}-2592 x^{{3}/{2}} y^{5} a_{6}-1944 x^{{5}/{2}} y a_{5}-2592 x^{3} y b_{4}-1296 x^{{3}/{2}} y^{4} a_{3}-2592 x^{2} y b_{2}+198 x^{6} y a_{3}+1944 x^{4} y a_{2}+216 x^{3} y^{2} a_{3}+1512 x^{3} y a_{1}-1944 x \,y^{2} a_{2}-x^{{27}/{2}} a_{3}-36 x^{9} a_{3}-324 x^{{9}/{2}} a_{3}+36 x^{7} b_{3}+432 x^{5} b_{2}+432 x^{4} b_{1}-1296 x^{{3}/{2}} y a_{3}-1296 x \,y^{2} b_{3}-2592 x y b_{1}-342 x^{9} a_{7}-648 y^{5} a_{10}-270 x^{7} a_{2}-234 x^{6} a_{1}-648 y^{3} a_{3}-648 y^{2} a_{1}-1944 x^{{5}/{2}} a_{2}+648 x^{{5}/{2}} b_{3}-1296 x^{{3}/{2}} a_{1}+1296 b_{2} \sqrt {x}-306 x^{8} a_{4}-648 y^{4} a_{6}+162 x^{7} y a_{5}+630 x^{6} y^{2} a_{6}+2376 x^{5} y a_{4}+648 x^{4} y^{2} a_{5}-1080 x^{3} y^{3} a_{6}-3240 x^{2} y^{2} a_{4}-1944 x \,y^{3} a_{5}+2592 x^{{3}/{2}} b_{4}+1296 y b_{5} \sqrt {x}+36 x^{8} b_{5}-2592 x^{{7}/{2}} a_{4}+648 x^{{7}/{2}} b_{5}+1296 x^{{5}/{2}} y b_{6}-x^{{29}/{2}} a_{5}-36 x^{10} a_{5}-324 x^{{11}/{2}} a_{5}-1296 x^{{3}/{2}} y^{2} a_{6}+432 x^{6} b_{4}+1944 x^{{5}/{2}} y^{2} b_{10}+1296 x \,y^{4} b_{10}-x^{{31}/{2}} a_{8}-36 x^{11} a_{8}-324 x^{{13}/{2}} a_{8}-3240 x^{{9}/{2}} a_{7}-1296 x^{{3}/{2}} y^{3} a_{10}+432 x^{7} b_{7}+3888 x^{{5}/{2}} b_{7}+1296 y^{2} b_{9} \sqrt {x}+36 x^{9} b_{8}+648 x^{{9}/{2}} b_{8}+126 x^{8} y a_{8}+594 x^{7} y^{2} a_{9}+1062 x^{6} y^{3} a_{10}+2808 x^{6} y a_{7}+1080 x^{5} y^{2} a_{8}-648 x^{4} y^{3} a_{9}-2376 x^{3} y^{4} a_{10}-4536 x^{3} y^{2} a_{7}-3240 x^{2} y^{3} a_{8}-1944 x \,y^{4} a_{9} = 0 \end{equation} Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them. \[ \left \{x, y, \sqrt {x}, x^{{3}/{2}}, x^{{5}/{2}}, x^{{7}/{2}}, x^{{9}/{2}}, x^{{11}/{2}}, x^{{13}/{2}}, x^{{15}/{2}}, x^{{17}/{2}}, x^{{19}/{2}}, x^{{21}/{2}}, x^{{23}/{2}}, x^{{25}/{2}}, x^{{27}/{2}}, x^{{29}/{2}}, x^{{31}/{2}}\right \} \] The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them \[ \left \{x = v_{1}, y = v_{2}, \sqrt {x} = v_{3}, x^{{3}/{2}} = v_{4}, x^{{5}/{2}} = v_{5}, x^{{7}/{2}} = v_{6}, x^{{9}/{2}} = v_{7}, x^{{11}/{2}} = v_{8}, x^{{13}/{2}} = v_{9}, x^{{15}/{2}} = v_{10}, x^{{17}/{2}} = v_{11}, x^{{19}/{2}} = v_{12}, x^{{21}/{2}} = v_{13}, x^{{23}/{2}} = v_{14}, x^{{25}/{2}} = v_{15}, x^{{27}/{2}} = v_{16}, x^{{29}/{2}} = v_{17}, x^{{31}/{2}} = v_{18}\right \} \] The above PDE (6E) now becomes \begin{equation} \tag{7E} 108 v_{1}^{7} v_{2}^{2} b_{10}-432 v_{1}^{5} v_{2}^{2} b_{9}-864 v_{1}^{4} v_{2}^{3} b_{10}-2592 v_{6} v_{2} a_{8}+1296 v_{6} v_{2} b_{9}+432 v_{1}^{5} b_{2}+432 v_{1}^{4} b_{1}-342 v_{1}^{9} a_{7}-648 v_{2}^{5} a_{10}-270 v_{1}^{7} a_{2}-234 v_{1}^{6} a_{1}-648 v_{2}^{3} a_{3}-648 v_{2}^{2} a_{1}-1944 v_{5} a_{2}+648 v_{5} b_{3}-1296 v_{4} a_{1}+1296 b_{2} v_{3}-306 v_{1}^{8} a_{4}-648 v_{2}^{4} a_{6}+2592 v_{4} b_{4}+36 v_{1}^{8} b_{5}-2592 v_{6} a_{4}+648 v_{6} b_{5}-v_{17} a_{5}-36 v_{1}^{10} a_{5}-324 v_{8} a_{5}+432 v_{1}^{6} b_{4}-v_{18} a_{8}-36 v_{1}^{11} a_{8}-324 v_{9} a_{8}-3240 v_{7} a_{7}+432 v_{1}^{7} b_{7}+3888 v_{5} b_{7}+36 v_{1}^{9} b_{8}+648 v_{7} b_{8}+864 v_{8} v_{2}^{3} a_{5}+1728 v_{7} v_{2}^{4} a_{6}-1296 v_{5} v_{2}^{4} a_{5}-2592 v_{4} v_{2}^{5} a_{6}-1944 v_{5} v_{2} a_{5}-2592 v_{1}^{3} v_{2} b_{4}-1296 v_{4} v_{2}^{4} a_{3}-2592 v_{1}^{2} v_{2} b_{2}+198 v_{1}^{6} v_{2} a_{3}+1944 v_{1}^{4} v_{2} a_{2}+216 v_{1}^{3} v_{2}^{2} a_{3}+1512 v_{1}^{3} v_{2} a_{1}-1944 v_{1} v_{2}^{2} a_{2}-1296 v_{4} v_{2} a_{3}-1296 v_{1} v_{2}^{2} b_{3}-2592 v_{1} v_{2} b_{1}+162 v_{1}^{7} v_{2} a_{5}+630 v_{1}^{6} v_{2}^{2} a_{6}+2376 v_{1}^{5} v_{2} a_{4}+648 v_{1}^{4} v_{2}^{2} a_{5}-1080 v_{1}^{3} v_{2}^{3} a_{6}-3240 v_{1}^{2} v_{2}^{2} a_{4}-1944 v_{1} v_{2}^{3} a_{5}+1296 v_{2} b_{5} v_{3}+1296 v_{5} v_{2} b_{6}-1296 v_{4} v_{2}^{2} a_{6}+1944 v_{5} v_{2}^{2} b_{10}+1296 v_{1} v_{2}^{4} b_{10}-1296 v_{4} v_{2}^{3} a_{10}+1296 v_{2}^{2} b_{9} v_{3}+126 v_{1}^{8} v_{2} a_{8}+594 v_{1}^{7} v_{2}^{2} a_{9}+1062 v_{1}^{6} v_{2}^{3} a_{10}+2808 v_{1}^{6} v_{2} a_{7}+1080 v_{1}^{5} v_{2}^{2} a_{8}-648 v_{1}^{4} v_{2}^{3} a_{9}-2376 v_{1}^{3} v_{2}^{4} a_{10}-v_{16} a_{3}-36 v_{1}^{9} a_{3}-324 v_{7} a_{3}+36 v_{1}^{7} b_{3}-1296 v_{1}^{3} v_{2}^{2} b_{8}-2 v_{17} v_{2} a_{9}-3 v_{16} v_{2}^{2} a_{10}+24 v_{15} v_{2} a_{8}+48 v_{14} v_{2}^{2} a_{9}+72 v_{13} v_{2}^{3} a_{10}-72 v_{1}^{10} v_{2} a_{9}-108 v_{1}^{9} v_{2}^{2} a_{10}-216 v_{12} v_{2}^{2} a_{8}-432 v_{11} v_{2}^{3} a_{9}-648 v_{10} v_{2}^{4} a_{10}-648 v_{8} v_{2} a_{9}-972 v_{7} v_{2}^{2} a_{10}+864 v_{9} v_{2}^{3} a_{8}+1728 v_{8} v_{2}^{4} a_{9}+2592 v_{7} v_{2}^{5} a_{10}-1296 v_{6} v_{2}^{4} a_{8}-2592 v_{5} v_{2}^{5} a_{9}-3888 v_{4} v_{2}^{6} a_{10}-1944 v_{5} v_{2}^{2} a_{9}-2592 v_{1}^{4} v_{2} b_{7}+2592 v_{4} v_{2} b_{8}+72 v_{1}^{8} v_{2} b_{9}+24 v_{13} v_{2} a_{3}-216 v_{10} v_{2}^{2} a_{3}+864 v_{7} v_{2}^{3} a_{3}+72 v_{1}^{7} v_{2} b_{6}-432 v_{1}^{4} v_{2}^{2} b_{6}-1296 v_{1}^{2} v_{2}^{2} b_{5}-2 v_{16} v_{2} a_{6}+24 v_{14} v_{2} a_{5}+48 v_{13} v_{2}^{2} a_{6}-72 v_{1}^{9} v_{2} a_{6}-216 v_{11} v_{2}^{2} a_{5}-432 v_{10} v_{2}^{3} a_{6}-648 v_{7} v_{2} a_{6}-4536 v_{1}^{3} v_{2}^{2} a_{7}-3240 v_{1}^{2} v_{2}^{3} a_{8}-1944 v_{1} v_{2}^{4} a_{9} = 0 \end{equation} Collecting the above on the terms \(v_i\) introduced, and these are \[ \{v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{7}, v_{8}, v_{9}, v_{10}, v_{11}, v_{12}, v_{13}, v_{14}, v_{15}, v_{16}, v_{17}, v_{18}\} \] Equation (7E) now becomes \begin{equation} \tag{8E} \left (126 a_{8}+72 b_{9}\right ) v_{2} v_{1}^{8}+\left (108 b_{10}+594 a_{9}\right ) v_{2}^{2} v_{1}^{7}+\left (-432 b_{9}+1080 a_{8}\right ) v_{2}^{2} v_{1}^{5}+\left (-864 b_{10}-648 a_{9}\right ) v_{2}^{3} v_{1}^{4}+\left (-2592 a_{8}+1296 b_{9}\right ) v_{2} v_{6}+\left (1296 b_{6}-1944 a_{5}\right ) v_{2} v_{5}+\left (-2592 b_{4}+1512 a_{1}\right ) v_{2} v_{1}^{3}+\left (198 a_{3}+2808 a_{7}\right ) v_{2} v_{1}^{6}+\left (1944 a_{2}-2592 b_{7}\right ) v_{2} v_{1}^{4}+\left (216 a_{3}-1296 b_{8}-4536 a_{7}\right ) v_{2}^{2} v_{1}^{3}+\left (-1944 a_{2}-1296 b_{3}\right ) v_{2}^{2} v_{1}+\left (-1296 a_{3}+2592 b_{8}\right ) v_{2} v_{4}+\left (72 b_{6}+162 a_{5}\right ) v_{2} v_{1}^{7}+\left (648 a_{5}-432 b_{6}\right ) v_{2}^{2} v_{1}^{4}+\left (-3240 a_{4}-1296 b_{5}\right ) v_{2}^{2} v_{1}^{2}+\left (1944 b_{10}-1944 a_{9}\right ) v_{2}^{2} v_{5}+\left (1296 b_{10}-1944 a_{9}\right ) v_{2}^{4} v_{1}+\left (-270 a_{2}+432 b_{7}+36 b_{3}\right ) v_{1}^{7}+\left (-342 a_{7}+36 b_{8}-36 a_{3}\right ) v_{1}^{9}+\left (-234 a_{1}+432 b_{4}\right ) v_{1}^{6}+\left (-306 a_{4}+36 b_{5}\right ) v_{1}^{8}+\left (-1296 a_{1}+2592 b_{4}\right ) v_{4}+\left (-1944 a_{2}+648 b_{3}+3888 b_{7}\right ) v_{5}+\left (-2592 a_{4}+648 b_{5}\right ) v_{6}+\left (-3240 a_{7}+648 b_{8}-324 a_{3}\right ) v_{7}+432 v_{1}^{5} b_{2}+432 v_{1}^{4} b_{1}-648 v_{2}^{5} a_{10}-648 v_{2}^{3} a_{3}-648 v_{2}^{2} a_{1}+1296 b_{2} v_{3}-648 v_{2}^{4} a_{6}-v_{17} a_{5}-36 v_{1}^{10} a_{5}-324 v_{8} a_{5}-v_{18} a_{8}-36 v_{1}^{11} a_{8}-324 v_{9} a_{8}+864 v_{8} v_{2}^{3} a_{5}+1728 v_{7} v_{2}^{4} a_{6}-1296 v_{5} v_{2}^{4} a_{5}-2592 v_{4} v_{2}^{5} a_{6}-1296 v_{4} v_{2}^{4} a_{3}-2592 v_{1}^{2} v_{2} b_{2}-2592 v_{1} v_{2} b_{1}+630 v_{1}^{6} v_{2}^{2} a_{6}+2376 v_{1}^{5} v_{2} a_{4}-1080 v_{1}^{3} v_{2}^{3} a_{6}-1944 v_{1} v_{2}^{3} a_{5}+1296 v_{2} b_{5} v_{3}-1296 v_{4} v_{2}^{2} a_{6}-1296 v_{4} v_{2}^{3} a_{10}+1296 v_{2}^{2} b_{9} v_{3}+1062 v_{1}^{6} v_{2}^{3} a_{10}-2376 v_{1}^{3} v_{2}^{4} a_{10}-v_{16} a_{3}-2 v_{17} v_{2} a_{9}-3 v_{16} v_{2}^{2} a_{10}+24 v_{15} v_{2} a_{8}+48 v_{14} v_{2}^{2} a_{9}+72 v_{13} v_{2}^{3} a_{10}-72 v_{1}^{10} v_{2} a_{9}-108 v_{1}^{9} v_{2}^{2} a_{10}-216 v_{12} v_{2}^{2} a_{8}-432 v_{11} v_{2}^{3} a_{9}-648 v_{10} v_{2}^{4} a_{10}-648 v_{8} v_{2} a_{9}-972 v_{7} v_{2}^{2} a_{10}+864 v_{9} v_{2}^{3} a_{8}+1728 v_{8} v_{2}^{4} a_{9}+2592 v_{7} v_{2}^{5} a_{10}-1296 v_{6} v_{2}^{4} a_{8}-2592 v_{5} v_{2}^{5} a_{9}-3888 v_{4} v_{2}^{6} a_{10}+24 v_{13} v_{2} a_{3}-216 v_{10} v_{2}^{2} a_{3}+864 v_{7} v_{2}^{3} a_{3}-2 v_{16} v_{2} a_{6}+24 v_{14} v_{2} a_{5}+48 v_{13} v_{2}^{2} a_{6}-72 v_{1}^{9} v_{2} a_{6}-216 v_{11} v_{2}^{2} a_{5}-432 v_{10} v_{2}^{3} a_{6}-648 v_{7} v_{2} a_{6}-3240 v_{1}^{2} v_{2}^{3} a_{8} = 0 \end{equation} Setting each coefficients in (8E) to zero gives the following equations to solve \begin {align*} -648 a_{1}&=0\\ -1296 a_{3}&=0\\ -648 a_{3}&=0\\ -216 a_{3}&=0\\ -a_{3}&=0\\ 24 a_{3}&=0\\ 864 a_{3}&=0\\ 2376 a_{4}&=0\\ -1944 a_{5}&=0\\ -1296 a_{5}&=0\\ -324 a_{5}&=0\\ -216 a_{5}&=0\\ -36 a_{5}&=0\\ -a_{5}&=0\\ 24 a_{5}&=0\\ 864 a_{5}&=0\\ -2592 a_{6}&=0\\ -1296 a_{6}&=0\\ -1080 a_{6}&=0\\ -648 a_{6}&=0\\ -432 a_{6}&=0\\ -72 a_{6}&=0\\ -2 a_{6}&=0\\ 48 a_{6}&=0\\ 630 a_{6}&=0\\ 1728 a_{6}&=0\\ -3240 a_{8}&=0\\ -1296 a_{8}&=0\\ -324 a_{8}&=0\\ -216 a_{8}&=0\\ -36 a_{8}&=0\\ -a_{8}&=0\\ 24 a_{8}&=0\\ 864 a_{8}&=0\\ -2592 a_{9}&=0\\ -648 a_{9}&=0\\ -432 a_{9}&=0\\ -72 a_{9}&=0\\ -2 a_{9}&=0\\ 48 a_{9}&=0\\ 1728 a_{9}&=0\\ -3888 a_{10}&=0\\ -2376 a_{10}&=0\\ -1296 a_{10}&=0\\ -972 a_{10}&=0\\ -648 a_{10}&=0\\ -108 a_{10}&=0\\ -3 a_{10}&=0\\ 72 a_{10}&=0\\ 1062 a_{10}&=0\\ 2592 a_{10}&=0\\ -2592 b_{1}&=0\\ 432 b_{1}&=0\\ -2592 b_{2}&=0\\ 432 b_{2}&=0\\ 1296 b_{2}&=0\\ 1296 b_{5}&=0\\ 1296 b_{9}&=0\\ -1296 a_{1}+2592 b_{4}&=0\\ -234 a_{1}+432 b_{4}&=0\\ -1944 a_{2}-1296 b_{3}&=0\\ 1944 a_{2}-2592 b_{7}&=0\\ -1296 a_{3}+2592 b_{8}&=0\\ 198 a_{3}+2808 a_{7}&=0\\ -3240 a_{4}-1296 b_{5}&=0\\ -2592 a_{4}+648 b_{5}&=0\\ -306 a_{4}+36 b_{5}&=0\\ 648 a_{5}-432 b_{6}&=0\\ -2592 a_{8}+1296 b_{9}&=0\\ 126 a_{8}+72 b_{9}&=0\\ -2592 b_{4}+1512 a_{1}&=0\\ 72 b_{6}+162 a_{5}&=0\\ 1296 b_{6}-1944 a_{5}&=0\\ -432 b_{9}+1080 a_{8}&=0\\ -864 b_{10}-648 a_{9}&=0\\ 108 b_{10}+594 a_{9}&=0\\ 1296 b_{10}-1944 a_{9}&=0\\ 1944 b_{10}-1944 a_{9}&=0\\ -1944 a_{2}+648 b_{3}+3888 b_{7}&=0\\ -270 a_{2}+432 b_{7}+36 b_{3}&=0\\ 216 a_{3}-1296 b_{8}-4536 a_{7}&=0\\ -3240 a_{7}+648 b_{8}-324 a_{3}&=0\\ -342 a_{7}+36 b_{8}-36 a_{3}&=0 \end {align*}
Solving the above equations for the unknowns gives \begin {align*} a_{1}&=0\\ a_{2}&=\frac {4 b_{7}}{3}\\ a_{3}&=0\\ a_{4}&=0\\ a_{5}&=0\\ a_{6}&=0\\ a_{7}&=0\\ a_{8}&=0\\ a_{9}&=0\\ a_{10}&=0\\ b_{1}&=0\\ b_{2}&=0\\ b_{3}&=-2 b_{7}\\ b_{4}&=0\\ b_{5}&=0\\ b_{6}&=0\\ b_{7}&=b_{7}\\ b_{8}&=0\\ b_{9}&=0\\ b_{10}&=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 &= \frac {4 x}{3} \\ \eta &= x^{3}-2 y \\ \end{align*} Shifting is now applied to make \(\xi =0\) in order to simplify the rest of the computation \begin {align*} \eta &= \eta - \omega \left (x,y\right ) \xi \\ &= x^{3}-2 y - \left (\frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y \,x^{3}+x^{6}\right ) \sqrt {x}}{36}\right ) \left (\frac {4 x}{3}\right ) \\ &= -\frac {x^{{15}/{2}}}{27}+\frac {4 x^{{9}/{2}} y}{9}+\frac {x^{3}}{3}-\frac {4 x^{{3}/{2}} y^{2}}{3}-2 y\\ \xi &= 0 \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 \(S(R)\). Since \(\xi =0\) then in this special case \begin {align*} R = x \end {align*}
\(S\) is found from \begin {align*} S &= \int { \frac {1}{\eta }} dy\\ &= \int { \frac {1}{-\frac {x^{{15}/{2}}}{27}+\frac {4 x^{{9}/{2}} y}{9}+\frac {x^{3}}{3}-\frac {4 x^{{3}/{2}} y^{2}}{3}-2 y}} dy \end {align*}
Which results in \begin {align*} S&= \operatorname {arctanh}\left (\frac {4 x^{{3}/{2}} y}{3}-\frac {2 x^{{9}/{2}}}{9}+1\right ) \end {align*}
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 {\left (18 x^{{3}/{2}}+36 y^{2}-12 y \,x^{3}+x^{6}\right ) \sqrt {x}}{36} \end {align*}
Evaluating all the partial derivatives gives \begin {align*} R_{x} &= 1\\ R_{y} &= 0\\ S_{x} &= \frac {-81 x^{3}+162 y}{36 x \left (x^{3}-4 x^{{3}/{2}} y^{2}+\frac {4 x^{{9}/{2}} y}{3}-6 y -\frac {x^{{15}/{2}}}{9}\right )}\\ S_{y} &= -\frac {27}{x^{{15}/{2}}-12 x^{{9}/{2}} y -9 x^{3}+36 x^{{3}/{2}} y^{2}+54 y} \end {align*}
Substituting all the above in (2) and simplifying gives the ode in canonical coordinates. \begin {align*} \frac {dS}{dR} &= -\frac {3}{4 x}\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}{4 R} \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 ) = -\frac {3 \ln \left (R \right )}{4}+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*} -\operatorname {arctanh}\left (-\frac {4 x^{{3}/{2}} y}{3}+\frac {2 x^{{9}/{2}}}{9}-1\right ) = -\frac {3 \ln \left (x \right )}{4}+c_{1} \end {align*}
Which simplifies to \begin {align*} -\operatorname {arctanh}\left (-\frac {4 x^{{3}/{2}} y}{3}+\frac {2 x^{{9}/{2}}}{9}-1\right ) = -\frac {3 \ln \left (x \right )}{4}+c_{1} \end {align*}
Which gives \begin {align*} y = \frac {2 x^{{9}/{2}}+9 \tanh \left (-\frac {3 \ln \left (x \right )}{4}+c_{1} \right )-9}{12 x^{{3}/{2}}} \end {align*}
The following diagram shows solution curves of the original ode and how they transform in the canonical coordinates space using the mapping shown.
|
Original ode in \(x,y\) coordinates |
Canonical coordinates transformation |
ODE in canonical coordinates \((R,S)\) |
| \( \frac {dy}{dx} = \frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y \,x^{3}+x^{6}\right ) \sqrt {x}}{36}\) |
|
\( \frac {d S}{d R} = -\frac {3}{4 R}\) |
| |
\(\!\begin {aligned} R&= x\\ S&= \operatorname {arctanh}\left (\frac {4 x^{{3}/{2}} y}{3}-\frac {2 x^{{9}/{2}}}{9}+1\right ) \end {aligned} \) |
|
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {2 x^{{9}/{2}}+9 \tanh \left (-\frac {3 \ln \left (x \right )}{4}+c_{1} \right )-9}{12 x^{{3}/{2}}} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 x^{{9}/{2}}+9 \tanh \left (-\frac {3 \ln \left (x \right )}{4}+c_{1} \right )-9}{12 x^{{3}/{2}}} \] Verified OK.
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y \,x^{3}+x^{6}\right ) \sqrt {x}}{36} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {x^{{13}/{2}}}{36}-\frac {x^{{7}/{2}} y}{3}+\frac {x^{2}}{2}+\sqrt {x}\, y^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {\left (x^{6}+18 x^{{3}/{2}}\right ) \sqrt {x}}{36}\), \(f_1(x)=-\frac {x^{{7}/{2}}}{3}\) and \(f_2(x)=\sqrt {x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\sqrt {x}\, u} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=\frac {1}{2 \sqrt {x}}\\ f_1 f_2 &=-\frac {x^{4}}{3}\\ f_2^2 f_0 &=\frac {x^{{3}/{2}} \left (x^{6}+18 x^{{3}/{2}}\right )}{36} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \sqrt {x}\, u^{\prime \prime }\left (x \right )-\left (\frac {1}{2 \sqrt {x}}-\frac {x^{4}}{3}\right ) u^{\prime }\left (x \right )+\frac {x^{{3}/{2}} \left (x^{6}+18 x^{{3}/{2}}\right ) u \left (x \right )}{36} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{-\frac {x^{{9}/{2}}}{27}} \left (c_{1} +x^{{3}/{2}} c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\left (c_{1} x^{{7}/{2}}+c_{2} \left (x^{5}-9 \sqrt {x}\right )\right ) {\mathrm e}^{-\frac {x^{{9}/{2}}}{27}}}{6} \] Using the above in (1) gives the solution \[ y = \frac {c_{1} x^{{7}/{2}}+c_{2} \left (x^{5}-9 \sqrt {x}\right )}{6 \sqrt {x}\, \left (c_{1} +x^{{3}/{2}} c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {c_{1} x^{{7}/{2}}+c_{2} \left (x^{5}-9 \sqrt {x}\right )}{\sqrt {x}\, \left (6 x^{{3}/{2}} c_{2} +6 c_{1} \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} x^{{7}/{2}}+c_{2} \left (x^{5}-9 \sqrt {x}\right )}{\sqrt {x}\, \left (6 x^{{3}/{2}} c_{2} +6 c_{1} \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1} x^{{7}/{2}}+c_{2} \left (x^{5}-9 \sqrt {x}\right )}{\sqrt {x}\, \left (6 x^{{3}/{2}} c_{2} +6 c_{1} \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y x^{3}+x^{6}\right ) \sqrt {x}}{36}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 y x^{3}+x^{6}\right ) \sqrt {x}}{36} \end {array} \]
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 trying exact Looking for potential symmetries trying Riccati trying Riccati sub-methods: <- Riccati particular case Kamke (d) successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 23
dsolve(diff(y(x),x) = 1/36*(18*x^(3/2)+36*y(x)^2-12*x^3*y(x)+x^6)*x^(1/2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{3}}{6}-\frac {3}{2 x^{\frac {3}{2}}-3 c_{1}} \]
✓ Solution by Mathematica
Time used: 0.199 (sec). Leaf size: 38
DSolve[y'[x] == (Sqrt[x]*(18*x^(3/2) + x^6 - 12*x^3*y[x] + 36*y[x]^2))/36,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^3}{6}+\frac {1}{-\frac {2 x^{3/2}}{3}+c_1} \\ y(x)\to \frac {x^3}{6} \\ \end{align*}