Internal problem ID [10164]
Internal file name [OUTPUT/9111_Monday_June_06_2022_06_40_43_AM_59121873/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1842.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying 3rd order ODE linearizable_by_differentiation differential order: 3; trying a linearization to 4th order trying differential order: 3; missing variables trying differential order: 3; exact nonlinear -> Calling odsolve with the ODE`, _a^2*(diff(diff(_b(_a), _a), _a))+(diff(_b(_a), _a))*_b(_a)*_a-3*(diff(_b(_a), _a))*_a-_b(_a)^2+4* 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 trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying symmetries linear in x and y(x) `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym Try integration with the canonical coordinates of the symmetry [_a, 0] -> Calling odsolve with the ODE`, diff(_g(_f), _f) = -_g(_f)^3*_f^2+_g(_f)^3*c__1+4*_g(_f)^3*_f+_g(_f)^2*_f-4*_g(_f)^2, _g(_f), e 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 trying Abel Looking for potential symmetries Looking for potential symmetries -> Calling odsolve with the ODE`, diff(y(x), x) = -(1/2)*y(x)^2+2*y(x)+(1/2)*(exp(-2*x)*c__1-2)/exp(-2*x), y(x), implicit` 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: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(u(x), x), x) = 2*(diff(u(x), x))+((1/4)*c__1-(1/2)*exp(2*x))*u(x), u(x)` Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler 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 <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful Change of variables used: [x = 1/2*ln(t)] Linear ODE actually solved: (-c__1+2*t)*u(t)+16*t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful <- Riccati to 2nd Order successful <- Abel successful <- differential order: 2; canonical coordinates successful <- differential order: 3; exact nonlinear successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 190
dsolve(x^2*diff(diff(diff(y(x),x),x),x)+x*(-1+y(x))*diff(diff(y(x),x),x)+x*diff(y(x),x)^2+(1-y(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ \ln \left (x \right )+2 \left (\int _{}^{y \left (x \right )}\frac {1}{2 \operatorname {RootOf}\left (-2 \sqrt {4+c_{1}}\, \operatorname {BesselY}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2} +2 \operatorname {BesselY}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2} \textit {\_h} -4 \operatorname {BesselY}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2} +2 \operatorname {BesselY}\left (\frac {\sqrt {4+c_{1}}}{2}+1, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {2}\, c_{2} \textit {\_Z} +2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_{1}}}{2}+1, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {2}\, \textit {\_Z} -2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {4+c_{1}}+2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \textit {\_h} -4 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right )\right )^{2}+\textit {\_h}^{2}-c_{1} -4 \textit {\_h}}d \textit {\_h} \right )-c_{3} = 0 \]
✓ Solution by Mathematica
Time used: 60.245 (sec). Leaf size: 282
DSolve[(1 - y[x])*y'[x] + x*y'[x]^2 + x*(-1 + y[x])*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {2 \left (c_3 \left (\operatorname {BesselJ}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}},-\frac {1}{2} i x \sqrt {c_1}\right )-\frac {1}{4} i \sqrt {c_1} x \left (\operatorname {BesselJ}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}}-1,-\frac {1}{2} i x \sqrt {c_1}\right )-\operatorname {BesselJ}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}}+1,-\frac {1}{2} i x \sqrt {c_1}\right )\right )\right )+\operatorname {BesselY}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}},-\frac {1}{2} i x \sqrt {c_1}\right )-\frac {1}{4} i \sqrt {c_1} x \left (\operatorname {BesselY}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}}-1,-\frac {1}{2} i x \sqrt {c_1}\right )-\operatorname {BesselY}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}}+1,-\frac {1}{2} i x \sqrt {c_1}\right )\right )\right )}{c_3 \operatorname {BesselJ}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}},-\frac {1}{2} i x \sqrt {c_1}\right )+\operatorname {BesselY}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}},-\frac {1}{2} i x \sqrt {c_1}\right )} \]