Section 24.1
Basic Concepts for n-th Order
Extending the theory from 2nd order to n-th order. Most properties generalize naturally.
1
Introduction
An -th order linear DE looks like:
We need initial conditions: .
2
Existence & Uniqueness
Theorem
If all and are continuous on an interval containing , then there exists a unique solution on .
3
Linear Independence
A set of functions is linearly independent if implies all .
The Wronskian is now an determinant of the functions and their derivatives up to .
4
Worked Examples
Example 1: Wronskian
Check if are independent.
Determinant of triangulation matrix = product of diagonal.
.
They are independent.
Example 2: Interval of Existence
with .
Standard form: .
Discontinuities at .
requires .
Initial condition at is in .
Unique solution exists on .
Example 3: Visual Basis
A fundamental set of solutions forms a "basis" for the solution space.
5
Practice Quiz
Loading...