How to solve this linear pde for y ( x ) (other functions are known and &#x03BB;<

herbariak1 2022-05-21 Answered
How to solve this linear pde for y ( x ) (other functions are known and λ is a constant):
d d x ( g ( x ) y ( x ) ) = λ 2 g ( x ) y ( x )
Everything I know about this equation is that it is called the Sturmian equation. I did some research, but the theory, which is for a more general form of the equation, is too hard for me to understand.
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Answers (1)

szilincsifs
Answered 2022-05-22 Author has 15 answers
Consider the operator
L f = 1 g d d x ( g d f d x ) .
You want to solve L f = λ 2 f. Start by solving for f 0 such that
L f 0 = 0 , f 0 ( 0 ) = A , f 0 ( 0 ) = B .
This is done by integrating the following in x:
d d x ( g d f 0 d x ) = 0 g d f 0 d x g ( 0 ) B = 0 d f 0 d x = B g ( 0 ) g ( x ) f 0 ( x ) = A + B 0 x g ( 0 ) g ( x ) d x .
Then recursively solve
1 g d d x ( g d f n d x ) = f n 1 , f n ( 0 ) = 0 , f n ( 0 ) = 0 g d f n d x = 0 x g ( y ) f n 1 ( y ) d y f n ( x ) = 0 x 1 g ( z ) 0 z g ( y ) f n 1 ( y ) d y d z
Now form the sum
f ( x , λ ) = n = 0 λ 2 n f n ( x ) .
You can verify that f ( 0 ) = f 0 ( 0 ) = A and f ( 0 ) = f 0 ( 0 ) = B. And,
L f = n = 1 λ 2 n f n 1 = λ 2 n = 0 λ 2 n f n = λ 2 f .
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