Question

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

Short answer

Answer:

Therefore, the summation notation can be written as

Step-by-step solution

Given Information

he given differential equation is:

We first calculate the derivative of the assumed power series solution

Use the Substitution method

Substitutingandinto the given differential equation;

Now, we need to shift the indices of the summation. When the indices of summation have the same starting point and the powers ofagreement, we combine the summations.

Since the given linear first order differential equation isfor allin some interval, we will have

As an identity and so we must haveor

Taking into consideration successive integer values of k.

By considering successive integer values of k starting with k=0, we find

and so on, whichis arbitrary.

Acquire a solution for the power series

Using the original assumed solution and the above results, we obtain a formal power series solution as

From the above step, we see that the pattern of the coefficient is

Where

Therefore, the summation notation can be written as: