Consider the th-order linear homogeneous differential equation y^ (n) + a_ n-1 (x)y^…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Ordinary differential equations

Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise

Problem

Consider the

n

th-order linear homogeneous differential equation

y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_1(x)y' + a_0(x)y = 0

on an interval

I

, with every coefficient

a_i

continuous on

I

. Using the existence and uniqueness theorem for the equivalent first-order system, prove that the set of solutions on

I

is a vector space of dimension exactly

n

(over

\mathbb{R}

if the

a_i

are real). State where continuity of the

a_i

on the whole of

I

is used.

Related theory

Ordinary differential equations

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