Question

In Problems, 35-38 proceed as in Example 4 and find the $ power series solution of the given linear first order differential equation.

Short answer

Answer:

Therefore, we can write the power series as follows

Step-by-step solution

Given Information

The differential equation is;

We assume that the given differential equation has a power series solution, which has the following form:

We aim to find the recurrence relation between the coefficients of the series. We, first, find the derivative of the power series as following

We shift the summation index toas the zeroth term of the power series equals to zero.

Use the Substitution method

We substitute (1) and (2) into the given differential equation, yields:

To create the summation index and the power of 

Our aim, now, is to make the summation index and the power offor the three series, in the same phase. We, first, have to assure that the first terms, for the three power series, are raised to the same power, which is not the case. Now, we take out the first term of the third power series, yields;

Now, we do the shift by replacing the summation index, for the first power series, byinstead of

Find the coefficients of the series in terms.

Which is equal to zero.

where

Implies,

and

Now, we have the recurrence relation. By using (3) and (4), We can find the coefficients of the series in terms of

Write the power series

From the pattern above, we can deduce a general formula for the coefficients of the power series.

Therefore, we can write the power series as follows: