Internal problem ID [5521]
Internal file name [OUTPUT/4769_Sunday_June_05_2022_03_05_23_PM_14949691/index.tex]
Book: Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section: Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises.
page 300
Problem number: 7.3.8 (b).
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.
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime }+y=0} \] 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. \[ x^{2} y^{\prime \prime }+y^{\prime }+y = 0 \] 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 {1}{x^{2}}\\ q(x) &= \frac {1}{x^{2}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([\infty ]\)
Irregular singular points : \([0]\)
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 successful <- special function solution successful`
✗ Solution by Maple
Order:=6; dsolve(x^2*diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 84
AsymptoticDSolveValue[x^2*y''[x]+y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 e^{\frac {1}{x}} \left (\frac {59241 x^5}{40}+\frac {1911 x^4}{8}+\frac {91 x^3}{2}+\frac {21 x^2}{2}+3 x+1\right ) x^2+c_1 \left (-\frac {91 x^5}{40}+\frac {7 x^4}{8}-\frac {x^3}{2}+\frac {x^2}{2}-x+1\right ) \]