Uni Practice Q

[u/Acrobatic_Badger_322]

I might just be silly it’s late and I’m ill but I just can’t get to the form required for the question. As a last resort I asked Ai and it said there was a typo in the question but I don’t know. Pls help!

[u/Jazzlike_Platypus430]

Substitute y = sum over k = 0 to infinity of a_k x^k and find y' and y'', differentiating term by term.

Then plug it into your equation and collect all the terms in front of x^k for any k. Since it's 0 identically, all coefficients must be zero. This should give you the recursive/difference equation for coefficients a_k.

[u/AnisiFructus]

Yeah, I got: a_{k+2} = \frac{-2}{k+2} a_{k+1} + \frac{-1}{(k+1)(k+2)} a_k

So I also think that the author probably swapped the coefficients of a_{k+1} and a_k.

In this (the original in your pic) form the answer should be: A=1/2 and B=1/(k+1)

But if the coefficients are swapped, then it is A=k+1 and B=-2. (Though in this case also the A and B could be swapped so A remain in the coefficient of a_{k+1} and B in the coef of a_k)

I would just complete it in the way it is now (it will probably say it's wrong if they really swaped the coefficients) and also write an email to the teacher, you still have 2 days until the deadline.

[u/Affectionate-Baby248 → u/AnisiFructus]

I agree, the coefficients were very likely swapped.

[u/Affectionate-Baby248]

Focus on the coefficient of x^k in y''+2y'+y for each k, and make sure it comes out to 0. The coefficients will come from x^(k+2), x^(k+1) and x^k, respectively, so you should get

(k+2)(k+1)a_{k+2}+2*(k+1)a_{k+1}+a_k=0.

Now solve for a_{k+2} by rearranging, which gives you

a_{k+2}=-2/(k+2) * a_{k+1}-1/(k+2)(k+1) * a_k.

Comparing the form, the answers are A=1/2 and B=-1/(k+1). The key here (which you and the AI were probably confused about) is that one of the answers is an expression in terms of k.

EDIT: It's very likely the question has a typo with a_k and a_{k+1} being accidentally swapped. In that case, the correct answers would be A=k+1 and B=-2.

[u/A-JackRobin-]

Find the greatest solution to the equation, so what does the equation describe