Question

In Problems, 3–6 find two power series solutions of the given differential equation about the ordinary point. Compare the series solutions with the solutions of the differential equations obtained using the method of Section 4.3. Try to explain any differences between the two forms of the solutions.

Short answer

Answer:

The solution is:

Step-by-step solution

Step 1:Given Information.

The given equation is:

Determining the two power series solution

We can always find two linearly independent solutions in the form of a power series centered at, that is, we can always find two linearly independent solutions in the form of a power series centered at.

For the given differential equation, we must find two linearly independent series solutions. The first and second derivatives of (1) are found as follows:

The first and second derivatives of this power series solution are calculated first.

The summation's indices must now be shifted. We combine the summations when the indices of summation have the same beginning point and the powers of x agree.

We shall have y"-y'=0 for all x in some interval because the stated linear first order differential equation is y"-y'=0.

As a result, we must have the following as an identity.

or

,

We discover this by considering successive integer values of k starting with k=0.

and so on, with any values forand.

We derive a formal power series solution by utilizing the original solution form and the aforementioned results.

We rewrite the solution as, here

Now, we know from the Maclaurin Series that the sumis the expansion if. As a result, we can simplify the solution once more as follows:

Let's use the method from section 4.3 to determine the solution:is the auxiliary equation of the equation y"-y'=0. The roots of this equation areandafter factoring and solving. As a result, the overall solution is;

It is the same as the general answer determined withandusing the method of section 4.3.