Internal problem ID [9848]
Internal file name [OUTPUT/8793_Monday_June_06_2022_05_29_54_AM_60943458/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1524.
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, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{6} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y=0} \] Unable to solve this ODE.
Maple trace
`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 trying Louvillian solutions for 3rd order ODEs, imprimitive case Louvillian solutions for 3rd order ODEs, imprimitive case: input is reducible, switching to DFactorsols checking if the LODE is of Euler type expon. solutions partially successful. Result(s) =`, [x^2]`Calling dsolve with: 2*x*y(x)+x^2*(2*x^3+1)*diff(y(x),x)+x^6*diff(diff(y( 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 -> elliptic -> Legendre <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 98
dsolve(x^6*diff(diff(diff(y(x),x),x),x)+x^2*diff(diff(y(x),x),x)-2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{2} \left (c_{1} +\left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{\frac {11}{2}}}d x \right ) c_{2} +\left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\operatorname {BesselK}\left (\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{\frac {11}{2}}}d x \right ) c_{3} \right ) \]
✓ Solution by Mathematica
Time used: 0.19 (sec). Leaf size: 96
DSolve[-2*y[x] + x^2*y''[x] + x^6*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {\left (-\frac {1}{3}\right )^{2/3} c_2 x \operatorname {Gamma}\left (\frac {1}{3}\right ) \, _2F_2\left (-\frac {2}{3},\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {1}{3 x^3}\right )}{3 \operatorname {Gamma}\left (\frac {4}{3}\right )}+\frac {c_3 \operatorname {Gamma}\left (\frac {2}{3}\right ) \, _2F_2\left (-\frac {1}{3},\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {1}{3 x^3}\right )}{9 \operatorname {Gamma}\left (\frac {5}{3}\right )}+c_1 x^2 \]