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In differential equations, the differentiable function or differentiation operator $L(y)$ can be defined (I ignore dimensions since its not the point here) acting $\mathbb R \longrightarrow \mathbb R$ or $\mathbb C \longrightarrow \mathbb R$ or $\mathbb R \longrightarrow \mathbb C$ or $\mathbb C \longrightarrow \mathbb C$ (am I right?).

What I don't understand is what to do with the different constants and parameters.

To be more specific, take for example: $$y''+ay'+y=e^{kx}$$

Its characteristic polynomial (for the homogenic solution) is $$r^2+a r+ 1=0$$

Its roots are: $$r_{1,2} = \frac{-a\pm \sqrt {a^2 - 4}} {2}$$ and the (homogenic) solution (if $r_1 \neq r_2$) has the form $$y(x) = c_1 e^{r_1x} + c_2 e^{r_2x}$$

So,

Considering the set $ PAR =$ { $a,k,r_1,r_2,c_1,c_2 $ }. Which elements of $PAR$ are allowed to be complex in each of the following four cases?

  1. $\mathbb R \longrightarrow \mathbb R$

  2. $\mathbb R \longrightarrow \mathbb C$

  3. $\mathbb C \longrightarrow \mathbb R$

  4. $\mathbb C \longrightarrow \mathbb C$

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  • $\begingroup$ I do not understand your first paragraph. $\endgroup$
    – GEdgar
    Jun 20, 2023 at 12:11

1 Answer 1

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I am not quite sure if I got your question right, but I'll try to shed some light on a few points.

First of all, a differential operator is a mapping from a function space to another function space. This could be for example $C^{\infty}(\mathbb{R})$ which is the space of infinitely many times differentiable functions.

So a differential operator "eats up" functions and "spits out" functions. Now as far as the parameters of an ODE are concerned, they can be either real or complex function in general. There isn't actually any rule which dictates where these parameters live.

However, based on whether the given coefficients are real or complex one usually deploys different methods in order to tackle the problem in question.

There are many excellent textbooks that present the theory of ODEs in quite an illustrative way.

If you are interested in the classical theory, then:

Ordinary Differential Equations by V. I. Arnol'd

is a great choice. Not the most easy to follow along but it definitely does a pretty good job of explaining the basic concepts. If you want to explore what happens when you try to solve an ODE on the complex plane then:

Ordinary Differential Equations by E. L. Ince

is the one to study.

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