Internal problem ID [1805]
Internal file name [OUTPUT/1806_Sunday_June_05_2022_02_32_53_AM_15775445/index.tex]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number: 13.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {t^{3} y^{\prime \prime }-t y^{\prime }-\left (t^{2}+\frac {5}{4}\right ) y=0} \] With the expansion point for the power series method at \(t = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ t^{3} y^{\prime \prime }-t y^{\prime }+\left (-t^{2}-\frac {5}{4}\right ) y = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(t) y^{\prime } + q(t) y &=0 \end {align*}
Where \begin {align*} p(t) &= -\frac {1}{t^{2}}\\ q(t) &= -\frac {4 t^{2}+5}{4 t^{3}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([]\)
Irregular singular points : \([0, \infty ]\)
Since \(t = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(t = 0\) is not regular singular point. Terminating.
Verification of solutions N/A
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- 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 -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> 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 a symmetry of the form [xi=0, eta=F(x)] trying differential order: 2; exact nonlinear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying to convert to an ODE of Bessel type trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
Order:=6; dsolve(t^3*diff(y(t),t$2)-t*diff(y(t),t)-(t^2+5/4)*y(t)=0,y(t),type='series',t=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 97
AsymptoticDSolveValue[t^3*y''[t]-t*y'[t]-(t^2+5/4)*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_2 e^{-1/t} \left (-\frac {239684276027 t^5}{8388608}+\frac {1648577803 t^4}{524288}-\frac {3127415 t^3}{8192}+\frac {26113 t^2}{512}-\frac {117 t}{16}+1\right ) t^{13/4}+\frac {c_1 \left (-\frac {784957 t^5}{8388608}-\frac {152693 t^4}{524288}-\frac {7649 t^3}{8192}-\frac {31 t^2}{512}+\frac {45 t}{16}+1\right )}{t^{5/4}} \]