Internal problem ID [6042]
Internal file name [OUTPUT/5290_Sunday_June_05_2022_03_29_41_PM_43044529/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number: 1(c).
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.
\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime }+3 y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ x^{2} y^{\prime \prime }-5 y^{\prime }+3 y x^{2} = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= -\frac {5}{x^{2}}\\ q(x) &= 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 \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) 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 <- Heun successful: received ODE is equivalent to the HeunD ODE, case c = 0 `
✗ Solution by Maple
Order:=8; dsolve(x^2*diff(y(x),x$2)-5*diff(y(x),x)+3*x^2*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 106
AsymptoticDSolveValue[x^2*y''[x]-5*y'[x]+3*x^2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {339 x^7}{8750}+\frac {49 x^6}{1250}+\frac {18 x^5}{625}+\frac {3 x^4}{50}+\frac {x^3}{5}+1\right )+c_2 e^{-5/x} \left (-\frac {302083 x^7}{218750}+\frac {5243 x^6}{6250}-\frac {357 x^5}{625}+\frac {113 x^4}{250}-\frac {49 x^3}{125}+\frac {6 x^2}{25}-\frac {2 x}{5}+1\right ) x^2 \]