Consider x y &#x2033; </msup> + 2 y &#x2032; </msup>

doodverft05

doodverft05

Answered question

2022-06-21

Consider x y + 2 y + x y = 0. Its solutions are cos x x , sin x x .
Neither of those solutions (as far as I could find) can be the solutions of a first order linear homogeneous differential equation. However
e ± i x x
Can be the solution of a first order linear homogeneous DE ( y + ( x i ) y = 0)
Can I always find a first order homogeneous linear DE whose solution also solves a second order homogeneous linear DE?
For example can I find a first order homogeneous linear DE whose solution is a particular linear combination of J 1 and Y 1 ?
(unrelated: also I'd like to know if there is a way of solving x y + 2 y + x y = 0 without noticing that it is a spherical bessel function or using laplace transform.)

Answer & Explanation

Abigail Palmer

Abigail Palmer

Beginner2022-06-22Added 30 answers

This may look like cheating, but any C 1 function ψ satiafies the first order differential equation
ψ + a ψ = 0 with a = ψ ψ .

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