Example 2 Find a series solution around x_(0)=0 for the following differential equation. y''-xy=0

Question

Answer 1

To find a series solution for the differential equation

around

, we can use the method of power series.Let's assume a power series solution of the form:

Taking the first and second derivatives, we get:

Substituting these expressions into the given differential equation, we have:

To simplify the equation, let's shift the indices of the series:

Now, we can combine the terms with the same power of

:

For the series to be equal to zero, each coefficient must be equal to zero:

Solving for

, we get:

This recursive relation allows us to find the coefficients of the power series solution. We can start with an initial value, such as

, and then use the relation to find the subsequent coefficients.For example, if we choose

, we can find the next coefficients as follows:

and so on.Therefore, the power series solution for the differential equation

around

is:

where the coefficients

are determined by the recursive relation

.

Soru