Internal problem ID [10098]
Internal file name [OUTPUT/9045_Monday_June_06_2022_06_16_44_AM_81131816/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1777 (book 6.186).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 y y^{\prime } x^{2}+3 x y^{2}=0} \]
Maple trace Kovacic algorithm successful
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation -> Calling odsolve with the ODE`, diff(diff(diff(y(x), x), x), x)-(3/8)*(-12*x^2*(diff(diff(y(x), x), x))-6*(diff(y(x), x))*x+y(x))/ Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 3; missing the dependent variable Equation is the 2nd symmetric power of diff(diff(y(x),x),x)+3/2*x^2/(x^3-1)*diff(y(x),x)-3/16*x/(x^3-1)*y(x) = 0 -> Attempting now to solve this lower order ODE trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- 2nd order ODE linearizable_by_differentiation successful`
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 53
dsolve(8*(-x^3+1)*y(x)*diff(diff(y(x),x),x)-4*(-x^3+1)*diff(y(x),x)^2-12*x^2*y(x)*diff(y(x),x)+3*x*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {{\left (\operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) c_{1} +\frac {c_{2} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{2}\right )}^{2} x}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 94.818 (sec). Leaf size: 708
DSolve[3*x*y[x]^2 - 12*x^2*y[x]*y'[x] - 4*(1 - x^3)*y'[x]^2 + 8*(1 - x^3)*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \exp \left (\int _1^x-\frac {-2 \sqrt {K[2]^2+K[2]+1} \sqrt {\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}} \left (4 K[2]^2+\left (\sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+4\right ) K[2]+\sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+4\right )+c_1 \sqrt {1-K[2]} \left (K[2]^2+K[2]+1\right ) \left (4 K[2]^2+\left (\sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+10\right ) K[2]+3 \sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+10\right )+\sqrt {1-K[2]} \left (K[2]^2+K[2]+1\right ) \left (4 K[2]^2+\left (\sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+10\right ) K[2]+3 \sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+10\right ) \int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]}{2 (1-K[2])^{3/2} \sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1} \left (K[2]^2+K[2]+1\right ) \left (\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}\right ) \left (c_1+\int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]\right )}dK[2]\right ) \]