Wronskian: W(f₁,...,fₙ) = det[fᵢ⁽ʲ⁻¹⁾(x)]. If W(x₀) ≠ 0 for some x₀, functions are linearly independent. For solutions of linear ODEs: W ≡ 0 or W never zero (Abel's theorem). W = ce^(−∫p(x)dx) for y''+p(x)y'+q(x)y=0.
How to Use
Select functions:
- Functions: Preset sets
- x: Evaluation point
- Output: W(x) value
Interpretation
W ≠ 0: linearly independent (conclusive). W ≡ 0: may or may not be dependent (inconclusive for general functions). For ODE solutions: W ≡ 0 ⟺ dependent (stronger result).
Abel's Theorem
For solutions of y''+p(x)y'+q(x)y=0: W(x) = W(x₀)exp(−∫p(t)dt). The Wronskian is either always zero or never zero. This connects the Wronskian to the ODE coefficients.
Step-by-Step Instructions
- 1Select function set.
- 2Enter x value.
- 3View derivative matrix.
- 4Get W(x).
- 5Test independence.