Which $n$th order differential equations have $n$ linearly independent solutions?

Question

In these notes (p. 28), it is stated that differential equation $28$ is a second order ordinary differential equation therefore there are two linearly independent solutions.

Which is the largest set $X$ of differential equations such that

$$\text{differential equation }x \in X \Rightarrow x \text{ is } n\text{th order and there are }n \text{ linearly independent solutions}?$$

I think that $X$ is quite likely to be a subset of all linear differential equations, but really I'm not sure. Ideally I'd be interested in a largest $X$, but I wouldn't mind is someone knew some fairly large subsets of $X$.

Answer 1

Equuation 28 has the form

(eq 1) c X'' + b X' + a X = K

where a, b, c, K are functions on x:

c=-h/ (2 pi 2 m)

b=0

a=U(x)

K=E

Given two solutions X, Y, the function Z = Y - X verifies the linear equation :

(eq 2) c Z'' + b Z' + a Z = 0

And inversely, given a solution X of (1), for any solution Z of (2), the function Y = X + Z is also a solution of (1).

So: the solution space of (1) is a translation of the solution space of (2).

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(2) is a linear homogeneus equation and we can easily check that :

1. The addition of two any solutions is also a solution
2. The multiplication of a solution by a constant is also a solution

So : The solution space of (1) is a vector space

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To check that the dimension is exactly 2, we transofrm the 2nd order differential equation (2) to a 1st order differential equation system by introducing W = Z':

(eq 3)

W ' = -b/c W - a/c Z

Z ' = W

In our case, this can be done since c≠0.

So we have a 2 dimension, 1st order differential equation system. Using the theorm of existence and unicity of ordinary differential equations, we have that :

Given an initial evaluation point x, for any inital values (z_0, w_0), there exists one, and only one, solution (Z,W) that verifies (3)

So we can build a bijective, linear map :

R² → {solutions of (3)}

QED