# How do you find a Power Series solution of a linear differential equation?

To find a solution of a linear ordinary differential equation

For the ordinary power series method, start by assuming the solution of the differential equation to be of the form

Computing the derivatives and applying them to the DE, we get:

Which is vallid if and only if

Therefore:

and

wich is the well known solution for this problem.

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To find a power series solution of a linear differential equation, follow these steps:

- Express the differential equation as a power series, substituting a power series for the unknown function.
- Differentiate the power series term by term to find expressions for derivatives of the unknown function.
- Substitute the power series and its derivatives into the original differential equation.
- Equate coefficients of like powers of the independent variable on both sides of the equation.
- Solve the resulting recurrence relations to find the coefficients of the power series.
- Write the power series solution using the found coefficients.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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