Find the First 4 Terms of Two Linear Solutions for y"+xy'-5y=0 at x=0

[qasker101]

Homework Statement

Find the first 4 terms of teach of the two linear independent solutions oftheEQUATION

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y"+xy'-5y=0 a the ordinary point x=0

I am stuck at the circled part of the question and unable to proceed. Any guidance and help is appreciated!

The Attempt at a Solution

I have attached an image with the part till which I got to circled in red. Please advise me on how to proceed! This is a problem my son is practicing for his exam for his DQ class. The attached scan is a copy of solution that his friend had who is unreachable at the moment.

 

[mighty2000]

Hello there,

Thank you for reaching out for help with this problem. I am more than happy to assist you in finding the first 4 terms of each of the two linear independent solutions for the given equation.

First, let's start by writing out the given equation in its standard form:

y'' + xy' - 5y = 0

Since the given point is an ordinary point (x=0), we can use the power series method to find the solutions. We will assume that the solutions can be written as a power series in x:

y = ∑n=0∞ anxn

Substituting this into the given equation, we get:

∑n=2∞ n(n-1)anxn-2 + ∑n=1∞ anxn+1 - 5∑n=0∞ anxn = 0

Now, we can group the terms according to their powers of x:

∑n=0∞ (n+2)(n+1)an+2xn + ∑n=0∞ (n+1)an+1xn - 5∑n=0∞ anxn = 0

Next, we can equate the coefficients of each power of x to 0, starting with x^0:

(n+2)(n+1)an+2 + (n+1)an+1 - 5an = 0

Simplifying this, we get:

(n+2)(n+6)an+2 = 0

Therefore, we have two possible values for n: n = -2 or n = -6. We will consider each of these cases separately.

Case 1: n = -2

Substituting n = -2 into the above equation, we get:

0a0 + 0a1 - 5a-2 = 0

Solving for a-2, we get:

a-2 = 0

Therefore, the coefficient of x^-2 is 0, which means that the solution cannot be written as a power series in x. We will have to consider a different value of n.

Case 2: n = -6

Substituting n = -6 into the above equation, we get:

0a0 + 0a1 - 5a-6 = 0

Solving for a-6, we get: