Can this method be applied to any ODE with a regular singular point at $x=0$?

  • Context: MHB 
  • Thread starter Thread starter oasi
  • Start date Start date
  • Tags Tags
    Ode Series
Click For Summary
SUMMARY

This discussion focuses on solving ordinary differential equations (ODEs) with a regular singular point at $x=0$ using the Method of Frobenius. Participants emphasize starting with the series representation $y=\sum\limits_{n=0}^{\infty}c_nx^n$ and taking derivatives to substitute into the differential equation. The method requires adjusting the series to $y=\sum_{n=0}^{\infty}c_{n}x^{n+r}$ to accommodate the singularity at $x=0$. This approach is definitive for handling ODEs with regular singular points.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with series solutions and power series
  • Knowledge of the Method of Frobenius
  • Basic calculus, including differentiation and summation
NEXT STEPS
  • Study the Method of Frobenius in detail
  • Explore examples of ODEs with regular singular points
  • Learn about convergence of power series solutions
  • Investigate the implications of singular points on solution behavior
USEFUL FOR

Mathematicians, physics students, and engineers dealing with differential equations, particularly those focusing on series solutions and singular point analysis.

Physics news on Phys.org
oasi said:
How can we solve this ODE with series

http://img841.imageshack.us/img841/8682/80858005.png

dwsmith said:
Start by letting $y=\sum\limits_{n=0}^{\infty}c_nx^n$ and taking the appropriate derivatives and plugging them into the DE.

Actually, since $x=0$ is a regular singular point of this DE, you're going to have to use the Method of Frobenius. Try
$$y=\sum_{n=0}^{\infty}c_{n}x^{n+r}.$$
 

Similar threads

Undergrad Converting Second Order ODE to Hypergeometric Function
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Solving an ODE with power series
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Spectral Theorem to Convert PDE into ODE
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Searching for radial part solution of Klein-Gordon type equation
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Can the ODE \psi''-y^2\psi=0 be solved using a general method?
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Rational Chebyshev Collocation Method For Damped Harmonic Oscilator
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Few questions about series solution of ODEs
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Characteristic curves for ##u_t + (1-2u)u_x = -1/4, u(x,0) = f(x)##
  • · Replies 2 ·
Replies
2
Views
3K
MHB What Methods Solve This Initial Value ODE?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Inverse ODE, Green's Functions, and series solution
  • · Replies 4 ·
Replies
4
Views
2K