Power series to solve 2nd order ordinary differential equations

• hbomb
In summary, the conversation is about finding a power series centered around a point and a specific example question. The link provided may be helpful, but the person has a question about summations with different starting indices and with x=1 instead of x=0.
hbomb
I need some help with power series.
I can't remember how to find a power series center around a point.
example question:
y"-xy'-y=0, x=1
I don't how to start this.

hbomb said:
I need some help with power series.
I can't remember how to find a power series center around a point.
example question:
y"-xy'-y=0, x=1
I don't how to start this.

You should find this link useful: http://tutorial.math.lamar.edu/AllBrowsers/3401/SeriesSolutions.asp" .

Last edited by a moderator:
Yes, thanks. The site has been helpful, but I have a question that I couldn't find an answer for. What happens if you have a summation with the starting index of the summation with n=0 but one of the summations you have an index of n=1. All the exponents are the same.

Also what happens if instead x=1?

If you start at n =1, you subtract 1 from the exponent. So:

$$\sum_{n=0}^{k} x^{n} = \sum_{n=1}^{k}x^{n-1}$$

1. What is a power series?

A power series is an infinite series of the form ∑n=0^∞ an(x-c)n, where an are the coefficients, x is the variable, and c is the center of the series. It is used to represent a function as a sum of terms with increasing powers of x.

2. How can power series be used to solve 2nd order ordinary differential equations?

By substituting the power series into the differential equation, we can find the recurrence relation between the coefficients. Solving this recurrence relation will give us the coefficients, which can be used to construct the power series solution to the differential equation.

3. What are the advantages of using power series to solve 2nd order ordinary differential equations?

Power series solutions can be used to approximate solutions to differential equations with high accuracy. They also provide a systematic way of finding solutions to differential equations, and can be used in cases where other methods may not be applicable.

4. Are there any limitations to using power series to solve 2nd order ordinary differential equations?

Power series solutions may not converge for all values of x, and in some cases, may only converge for a limited interval. Additionally, finding the recurrence relation and solving it to obtain the coefficients can be a tedious and time-consuming process.

5. Can power series be used to solve higher order differential equations?

Yes, power series can be used to solve higher order differential equations. However, as the order of the differential equation increases, the recurrence relation becomes more complex and the solution may only converge for a smaller interval or may not converge at all.

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