y′-y=e^x y^2 s a simple differential equation, and I would solve it with a simple change of the variable, however, I was wondering if I could use Laplace transform to make it easier and more direct. There a way to do ccL([y(x)]^2)?

bamakhosimz

bamakhosimz

Answered question

2022-09-06

I was solving the differential equation
y y = e x y 2
I know that this is a simple differential equation, and I would solve it with a simple change of the variable, however, I was wondering if I could use Laplace transform to make it easier and more direct.
So is there a way to do L ( [ y ( x ) ] 2 )?

Answer & Explanation

Waylon Jenkins

Waylon Jenkins

Beginner2022-09-07Added 17 answers

y   y = e x y 2 
Substitute u =  1 y then solve with Laplace transform:
u  + u = e x 
Laplace transform cannot be used with non-linear DE otherwise.

obojeneqk

obojeneqk

Beginner2022-09-08Added 3 answers

Assume that y is analytical at 0 and has a power series.
y ( x ) =  k = 0  a k x k .
Then
y 2 ( x ) =  n = 0  (  i = 0 n a i a n  i ) x n 
and
L { y 2 } ( s ) =  n = 0  (  i = 0 n a i a n  i ) n ! s n + 1 ,
wherever that series converges (if it converges anywhere).
Undoubtedly, one can extend beyond 0; however, this does not make the problem any better.

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