Frobenius Method of Solving ODE

In summary, the problem asks for finding two series solutions for the given differential equation with a regular point at x = 0. The solution is in the form of a sum from n = 0 to infinity, with coefficients cn and powers of x. The attempt at solving the problem involves substituting the derivatives into the equation and combining the sums to start at the same index. However, determining the indicial equation and the corresponding recursive formulas have proved to be difficult. Setting the coefficients of the x-1 term to 0 results in two values for r, but the recursive formulas lead to four series solutions instead of the required two. Any help or suggestions would be appreciated.
  • #1
IImattII
1
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Homework Statement



Find two series solutions of the given DE about the regular point x = 0.
2xy'' + 5y' + xy = 0

Homework Equations



The answer is in the form y = SUM(cnxn+r) where SUM is the sum from n = 0 to infinity.

The Attempt at a Solution



This questions got me stumped. I'm able to substitute the second and first order derivatives into the equation. I'm also able to to combine the different sums so they start at the same index. The equation I end up with is:

xr[r(2r+3)c0x-1 + [2r(r+1)+5(r+1)]c1 + SUM[(k+r+1)(2k+2r+5)ck+1 + ck-1]xk] = 0
where in this case, SUM is the sum from k = 1 to infinity

I guess my problem is determining the indicial equation and thus the recursive formulas. I've tried setting the coefficients of the x-1 term equal to 0 and using that as my indicial equation resulting in r = 0 and -3/2 but the two resulting recursive formulas end in a solution with two series each so instead of only two series solutions I have four.
 
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  • #2
I'm assuming I'm just not setting up the problem correctly. Any help/suggestions would be greatly appreciated!
 

FAQ: Frobenius Method of Solving ODE

What is the Frobenius method of solving ODE?

The Frobenius method is a technique used to find a series solution to a second-order ordinary differential equation (ODE) with a regular singular point. It is based on the idea of representing the solution as a power series, which is then substituted into the ODE to determine the coefficients of the series.

When is the Frobenius method applicable?

The Frobenius method is applicable when the ODE has a regular singular point, which is a point where the coefficient of the highest-order derivative term is either zero or undefined. This method is particularly useful for solving ODEs that arise in physics and engineering problems.

What is a regular singular point?

A regular singular point is a point in the domain of a differential equation where the equation cannot be written in standard form. This means that the coefficient of the highest-order derivative term is either zero or undefined at this point.

How is the Frobenius method different from the power series method?

The Frobenius method is a specific type of power series method that is used for solving ODEs with regular singular points. Unlike the general power series method, the Frobenius method allows for the presence of terms with fractional powers in the power series solution, which is necessary for solving ODEs with regular singular points.

Can the Frobenius method be used to solve all types of ODEs?

No, the Frobenius method can only be used to solve ODEs with regular singular points. It cannot be applied to ODEs with irregular singular points or non-singular points. In these cases, other methods such as the Laplace transform or separation of variables must be used.

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