Internal problem ID [12405]
Internal file name [OUTPUT/11058_Wednesday_October_04_2023_07_06_03_PM_64152595/index.tex]
Book: Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham,
S.R.Otto. Cambridge Univ. Press 2003
Section: Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL
EQUATIONS. Problems page 28
Problem number: Problem 1.11(a).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_bessel_ode", "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_Emden, _Fowler]]
Unable to solve or complete the solution.
\[ \boxed {x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y=0} \] 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. \[ x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0 \] 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}\\ q(x) &= \frac {1}{x^{3}}\\ \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 <- Bessel successful <- special function solution successful`
✗ Solution by Maple
Order:=6; dsolve(x^3*diff(y(x),x$2)+x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.057 (sec). Leaf size: 222
AsymptoticDSolveValue[x^3*y''[x]+x^2*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} \sqrt [4]{x} \left (\frac {418854310875 i x^{9/2}}{8796093022208}-\frac {57972915 i x^{7/2}}{4294967296}+\frac {59535 i x^{5/2}}{8388608}-\frac {75 i x^{3/2}}{8192}-\frac {30241281245175 x^5}{281474976710656}+\frac {13043905875 x^4}{549755813888}-\frac {2401245 x^3}{268435456}+\frac {3675 x^2}{524288}-\frac {9 x}{512}+\frac {i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} \sqrt [4]{x} \left (-\frac {418854310875 i x^{9/2}}{8796093022208}+\frac {57972915 i x^{7/2}}{4294967296}-\frac {59535 i x^{5/2}}{8388608}+\frac {75 i x^{3/2}}{8192}-\frac {30241281245175 x^5}{281474976710656}+\frac {13043905875 x^4}{549755813888}-\frac {2401245 x^3}{268435456}+\frac {3675 x^2}{524288}-\frac {9 x}{512}-\frac {i \sqrt {x}}{16}+1\right ) \]