Question

In Problems, 3–6 find two power series solutions of the given differential equation about the ordinary point x 5 0. 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:

Therefore, the solution is:

Step-by-step solution

Given Information.

The given equation is:

Determining the two power series solution.

We've got

At, it has an ordinary point. Without providing proof, we assert that if a differential equation has an ordinary point at, it has a power series solution with two linearly independent solutions of the form

As a result, the following differential equation finds a solution:

Where

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

We change y's summation index to n=1 because the power series' first term is Zero and second term n=2 for y”.

(1), (2) and (3) are substituted into the above differential equation to produce

Our current goal is to combine the summation index and the power of x for the two series. To begin, we must ensure that the first terms of both power series are raised to the same power, which they are. Now, for the first power series, we make the shift by changing the summation index from n to n+1. It's worth noting that this does not affect the series terms; the shift gives identical results.

It suggests that;

Now, we'll use the recurrence relation to find the series' coefficients.

As a result, we can get a general equation for the series' coefficients. For n=2, 3,....,. Now, if we replace the result in (1), we get

Obtained using the Power Series Method, which is the universal solution for the given differential equation. Section 4.3 Methodology.

Using the section 4.3 method

To solve the given differential equation, we now employ the procedure described in section 4.3. Assume we have a differential equation that looks like this.

This means that if you take the second derivative of, you'll receive the first derivative multiplied by a constant, which characterizes the exponential function. As a result, we assume that the answer takes the following form:

When we plug into the given differential equation, we get

The only way to kill the given differential equation is to use the exponential function's coefficient, which provides.

When we solve for, we get

and

Then there are two options.

Using the Maclaurin series

Using the Maclaurin series for the exponential function, we find

A, C,and B are all arbitrary constants that can have any value. When you compare (4) and (5), you'll notice that they have the same shape.