Wronskian Calculator | Linear Independence & Fundamental Solution Sets
Compute the Wronskian determinant W(f₁, f₂) or W(f₁, f₂, f₃) by evaluating derivatives numerically. Tests linear independence of function sets, identifies fundamental solution sets for linear ODEs, and displays the full derivative matrix evaluated at multiple points.
Wronskian Calculator
Supported: sin, cos, tan, exp, sqrt, log, abs, ^, *, +, -, /, (, )
What Is the Wronskian Calculator | Linear Independence & Fundamental Solution Sets?
The Wronskian W(f₁, …, fₙ) is the determinant of the matrix whose rows are a set of functions and their successive derivatives. It is primarily used to test linear independence: if W(x) ≠ 0 for at least one x, the functions are linearly independent. If W(x) = 0 for all x, the functions may be linearly dependent (though W = 0 everywhere is necessary but not sufficient for dependence). This calculator uses numerical differentiation (central differences with h = 10⁻⁵) to evaluate W at five test points x = −2, −1, 0, 1, 2.
Formula
W(f₁,f₂) = f₁·f₂′ − f₂·f₁′ | W(f₁,f₂,f₃) = det [[f₁,f₂,f₃],[f₁′,f₂′,f₃′],[f₁″,f₂″,f₃″]]
How to Use
- 1
Choose "2 Functions" or "3 Functions" mode using the toggle buttons.
- 2
Click a preset (e.g. {sin(x), cos(x)}) to pre-fill the function fields, or type your own expressions.
- 3
Write expressions using: sin, cos, tan, exp, log, sqrt, abs, x, +, −, *, /, ^ (power), and parentheses.
- 4
Click "Compute Wronskian" — results appear in a table evaluated at x = −2, −1, 0, 1, 2.
- 5
The linear independence verdict is shown at the top: INDEPENDENT (W ≠ 0 at some point) or likely DEPENDENT (W ≈ 0 everywhere).
- 6
Each table row shows f₁(x), f₂(x), their derivatives, and W(x) — nonzero W values are highlighted.
- 7
For 3 functions, the second derivatives and the full 3×3 determinant are computed and displayed.
Select 2 or 3 function mode, enter expressions using standard math notation, and click Compute Wronskian.
Example Calculation
For f₁ = sin(x) and f₂ = cos(x): f₁′ = cos(x), f₂′ = −sin(x). W = sin(x)·(−sin(x)) − cos(x)·cos(x) = −sin²(x) − cos²(x) = −1 ≠ 0. The functions are linearly independent (they form a fundamental solution set for y″ + y = 0). For f₁ = x and f₂ = 2x: f₁′ = 1, f₂′ = 2. W = x·2 − 2x·1 = 0. Linearly dependent (f₂ = 2f₁).
Understanding Wronskian | Linear Independence & Fundamental Solution Sets
Common Function Pairs and Their Wronskians
| f₁(x) | f₂(x) | W(f₁, f₂) | Independent? | ODE Source |
|---|---|---|---|---|
| sin(x) | cos(x) | −1 | Yes | y'' + y = 0 |
| eˣ | e²ˣ | e³ˣ | Yes | y'' − 3y' + 2y = 0 |
| eˣ | x·eˣ | e²ˣ | Yes | y'' − 2y' + y = 0 (repeated root) |
| x | 2x | 0 | No | f₂ = 2f₁ |
| x² | x³ | x⁴ | Yes | Euler–Cauchy equation |
| 1 | x | 1 | Yes | y'' = 0 |
| sin²(x) | 1 − cos²(x) | 0 | No | Pythagorean identity |
| eˣ·sin(x) | eˣ·cos(x) | −e²ˣ | Yes | y'' − 2y' + 2y = 0 |
3-Function Wronskian Examples
| f₁ | f₂ | f₃ | W(f₁,f₂,f₃) | Independent? |
|---|---|---|---|---|
| 1 | x | x² | 2 | Yes |
| eˣ | e²ˣ | e³ˣ | 2e⁶ˣ | Yes |
| sin(x) | cos(x) | x | non-zero | Yes |
| x | x² | x³ | 2x³ | Yes (for x≠0) |
| 1 | 2 | 3 | 0 | No (constants) |
| eˣ | xeˣ | x²eˣ | 2e³ˣ | Yes |
Key Theorems About the Wronskian
- ›Abel's Theorem: For two solutions of y″ + p(x)y′ + q(x)y = 0, W(x) = C·e^(−∫p(x)dx). The Wronskian is either identically zero or never zero.
- ›Liouville's Formula (general): For an n-th order ODE, d/dx[W] = −tr(A)·W where A is the companion matrix — a generalization of Abel's theorem.
- ›Independence Criterion: W(x₀) ≠ 0 for some x₀ implies the functions are linearly independent. The converse fails in general (only holds for ODE solutions on intervals with continuous coefficients).
- ›Fundamental Solution Set: n solutions of an n-th order linear ODE form a fundamental set iff their Wronskian is nonzero at a single point of the domain.
- ›Variation of Parameters: The particular solution of y″ + py′ + qy = f is built using the Wronskian W = f₁f₂′ − f₂f₁′ through the integrals u₁ = −∫f·f₂/W dx and u₂ = ∫f·f₁/W dx.
Frequently Asked Questions
What does the Wronskian tell us about solutions to a differential equation?
For a second-order linear ODE y″ + p(x)y′ + q(x)y = 0, two solutions f₁ and f₂ form a fundamental solution set (i.e., the general solution is c₁f₁ + c₂f₂) if and only if their Wronskian W(f₁, f₂) is nonzero on the interval of interest. A zero Wronskian at any point in an open interval implies W = 0 everywhere (Abel's theorem).
Is W = 0 everywhere the same as linear dependence?
Not in general. Linear dependence always implies W = 0. But W = 0 everywhere does NOT always imply linear dependence — there exist pathological examples of linearly independent functions whose Wronskian vanishes identically. However, for solutions of a linear ODE with continuous coefficients on an interval, W = 0 at one point iff linear dependence.
Why does the calculator use numerical derivatives?
Symbolic differentiation requires a computer algebra system (CAS). This calculator evaluates expressions numerically via the Function() constructor and uses the central difference formula f′(x) ≈ (f(x+h) − f(x−h))/(2h) with h = 10⁻⁵, which is accurate to about 10 decimal places for smooth functions.
What is Abel's identity?
Abel's identity states that for two solutions f₁, f₂ of y″ + p(x)y′ + q(x)y = 0, the Wronskian satisfies W(x) = W(x₀)·exp(−∫p(t)dt from x₀ to x). This means W is either identically zero or never zero on the interval — it cannot be zero at one point and nonzero at another.
What expressions are supported by the calculator?
The expression parser supports: arithmetic operators +, −, *, / and ^ (power); functions sin, cos, tan, exp, log (natural), sqrt, abs; the variable x; and parentheses. Use explicit multiplication: write 2*x not 2x. Constants: enter pi as 3.14159 or use Math.PI-style input won't work — just approximate.
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