More on the Wronskian
Abel's Theorem was clean. But for random functions that aren't solutions to a nice ODE, the Wronskian can be tricky.
Introduction
Review: Two functions are Linearly Dependent if for all (constants not both zero).
If they are independent, they are "different functions".
The Caveat
For Solutions
If solve :
Dependent.
Independent.
For General Functions
Independent.
Dependent!
(Example: and ).
Visual: Peano's Functions
Interactive: Subtle Independence
For , . Same for . everywhere, yet you cannot write for all .
Worked Examples
Example 1: Wronskian of Polynomials
Calculate . (Independence of 3 functions requires a determinant).
1. Matrix:
2. Expand:
Use cofactor expansion along first column.
.
.
3. Conclusion:
Since (except at ), they are linearly independent.
Example 2: Reverse Engineering
Find the DE for which and are solutions.
1. Calculate Derivatives:
.
.
2. Set up System:
Plug into :
(A) .
(B) .
3. Solve System:
Subtract (A) from (B): .
From (A): .
4. Result:
.
Example 3: Wronskian of Trig Functions
Check independence of and .
1. Calculate Wronskian?
Wait, simplify first!
.
2. Compare:
.
Since one is a constant multiple of the other, they are Linearly Dependent.
3. Verify with Wronskian:
.
.
Consistent.
Practice Quiz
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