Series solution for differential equation

• Sam2000009
In summary, the conversation discusses obtaining a series solution for the given differential equation, with one approach being to assume a power series for y, differentiate it, and combine the resulting series into one. However, the speaker is struggling with grouping the series together and asks for help.
Sam2000009
<OP warned about not using the homework template>

Obtain a series solution of the diﬀerential equation x(x − 1)y" + [5x − 1]y' + 4y = 0Do I start by solving it normally then getting a series for the solution or assume y=power series differentiate then add up the series?

I did the latter and got three different but (slightly similar) looking series and the problem asks to group them all into one series which I cannot do

Last edited by a moderator:
Sam2000009 said:
I did the latter and got three different but (slightly similar) looking series and the problem asks to group them all into one series which I cannot do
You have a series for y, e.g. ##y = \displaystyle \sum_{n=0}^\infty c_nx^n## so that $$y' = \sum_{n=1}^\infty n c_nx^{n-1}\quad{\rm {and}}\quad y'' = \sum_{n=2}^\infty n (n-1) c_nx^{n-2} \ ,$$ right ?

Now work out the coefficient of ##\ x^n ## in x(x − 1)y" + [5x − 1]y' + 4y = 0

Last edited:
Sam2000009 said:
The problem asks to group them all into one series which I cannot do.
What's stopping you?

1. What is a series solution for a differential equation?

A series solution for a differential equation is a method used to approximate the solution of a differential equation as an infinite sum of simpler functions. This allows for a more manageable solution to be found, especially for complex or non-linear differential equations.

2. How is a series solution obtained?

A series solution is obtained by assuming that the solution to the differential equation can be expressed as a power series, where each term in the series is a polynomial function. This series is then substituted into the original differential equation and the coefficients of the terms are determined by solving for each term.

3. When is a series solution useful?

A series solution is useful when the differential equation cannot be solved using traditional methods, such as separation of variables or substitution. It is also useful when the solution is too complex to be expressed in a closed form.

4. What are the advantages of using a series solution?

One advantage of using a series solution is that it allows for a more accurate approximation of the solution compared to other methods. Additionally, it can be used to solve a wider range of differential equations, including those with non-constant coefficients or non-polynomial terms.

5. Are there any limitations to using a series solution?

Yes, there are limitations to using a series solution. It may not always be possible to find a series solution for a given differential equation, and even when a series solution can be obtained, it may not converge to the true solution for all values of the independent variable. It is important to check for convergence when using a series solution.

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