Find y which satisfies: y'=y^{a}, y(a)=a-2, for a \in N

Michael Maggard

Michael Maggard

Answered question

2022-01-20

Find y which satisfies:
y=ya,y(a)=a2, for aN

Answer & Explanation

Serita Dewitt

Serita Dewitt

Beginner2022-01-20Added 41 answers

This is what I did:
yyadx=1dx
ya+1a+1+c1=x+c2
ya+1=(x+C)(a+1),
where C=c2c1, and for a=1 there's no solution.
So I get y=(1(x+c)(1a))a1,
finding c is not pleasant.
I assume that something is wrong, Am I suppose leave y in the way that I find it after finding c?
Alex Sheppard

Alex Sheppard

Beginner2022-01-21Added 36 answers

I presume for each a your need to find one function ya.
Your working seems essentially correct, but you can re-write as
1(1a)ya1x=C
Plug in the required values of x and y and find C.
RizerMix

RizerMix

Expert2022-01-27Added 583 answers

There is nothing wrong, except abandoning poor a=1, which though a little special gives no problems. When we do the details we will see there is a problem at a=2. Quite quickly (for a1) we reach ya+1a+1=x+C. It is best to find C now. Put x=a. We get (a2)a+1a+1=a+C Now we know C, except when a=2 (one cannot divide by 0). So for a=2 there is no solution that satisfies the initial condition. There is no trouble if a=1. True, the above general formula does not quite work. But if we integrate, we get ln(|y|)=x+C. Put x=1. We can now solve for C, and end up with y=ex1. Alternately, we end up with \(y=Ce^{x}),

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