I am studying about the linear odes with non-constant coefficients. I know the first order linear o

tinydancer27br

tinydancer27br

Answered question

2022-05-18

I am studying about the linear odes with non-constant coefficients.
I know the first order linear ode with non-constant coefficient
(1) y ( x ) + f ( x ) y ( x ) = 0
has a general solution of the form
(2) y = C e f ( x ) d x
However, I am more interested in the case of linear second order odes with non-constant coefficients
(3) y ( x ) + g ( x ) y ( x ) + f ( x ) y ( x ) = 0
I know that this equation does not have a closed form solution like (2). However, I am interested in special cases of that.
Questions
1. Consider (3), when g ( x ) = 0, then we have
(4) y ( x ) + f ( x ) y ( x ) = 0
Is Eq.(4) a famous well-known equation? If YES, what is its name?
2. Does (4) have a closed form solution like (2)?
3. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?
For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

Answer & Explanation

Ronnie Glenn

Ronnie Glenn

Beginner2022-05-19Added 11 answers

Somewhere (but where ?) I already answer to a question quite the same as your question 1 (only change a sign in equation 4). By luck, I didn't remove my draft (copy below).
There is no general formula for the solutions of equation 4 in cases of any f(x) since specific special functions are defined according to each specific case. All the more so for equation 3.
COPY :
y ( x ) = f ( x ) y ( x )
Case:
f ( x ) = c 2 y ( x ) = c 1 e c x + c 2 e c x = c 3 cosh ( c x ) + c 4 sinh ( c x )
Case: f ( x ) = c 2 y ( x ) = c 1 cos ( c x ) + c 2 sin ( c x )
Case: f ( x ) = x y ( x ) = c 1 A i ( x ) + c 2 B i ( x ) . Ary functions.
Case: f ( x ) = x 2 y ( x ) = c 1 D 1 / 2 ( 2 x ) + c 2 D 1 / 2 ( i 2 x ) . Parabolic cylinder function.
Case: f ( x ) = λ 2 x 1 ν 2 y ( x ) = c 1 x J ν ( λ x ) + c 2 x Y ν ( λ x ) . Bessel functions
Case: f ( x ) = λ 2 x 1 ν 2 y ( x ) = c 1 x I ν ( λ x ) + c 2 x K ν ( λ x ) . Modified Bessel functions.
Case: f ( x ) = a + 2 b cos ( 2 x ) y ( x ) = c 1 C ( a , b ; x ) + c 1 S ( a , b ; x ). Mathieu functions.
Case: f ( x ) = A x 2 + B x + C y ( x ) = e γ 2 x x β 2 ( c 1 M ( α , β ; γ x ) + c 2 U ( α , β ; γ x ) ) with
f ( x ) = A x 2 + B x + C y ( x ) = e γ 2 x x β 2 ( c 1 M ( α , β ; γ x ) + c 2 U ( α , β ; γ x ) )
{ γ = ± 2 C β = 1 ± 2 A + 1 4 α = β 2 + B γ
Kummer and Tricomi functions (confluent hypergeometric functions).
Etc.

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