What is the Wronskian?
Type 2 to 4 functions of one variable, separated by commas, such as e^x, e^(-x). The page builds the Wronskian determinant and says what it shows about linear independence.
- Wronskian W
- -2
The Wronskian of e^x, e^(-x) is -2.
- Linear independence
- Linearly independent: W is not 0 everywhere.
Wronskian W: -2. The Wronskian of e^x, e^(-x) is -2.
How to calculate
Finds the Wronskian determinant of 2 to 4 functions of one variable and whether it shows linear independence, checked numerically.
Example with the default inputs (Functions e^x, e^(-x)): The Wronskian of e^x, e^(-x) is -2.
Method: W is the determinant whose rows are the functions and their 1st to (n − 1)th derivatives, each derivative checked against difference quotients.
- Radians. An answer that fails its check is not shown.
Worked examples
Each example is checked against the calculator on every build.
- Functions e^x, e^(-x) gives Wronskian W -2.Source: Trench (2013), Elementary Differential Equations, 5.1, Example 5.1.5 (a)
- Functions x^2, 1/x^2 gives Wronskian W -4/x.Source: Trench (2013), Elementary Differential Equations, 5.1, Example 5.1.5 (c)
How it works
For n functions y₁, …, yₙ of one letter (n = 2, 3 or 4), the Wronskian is the determinant
W = det [ y₁ … yₙ ; y₁′ … yₙ′ ; … ; y₁⁽ⁿ⁻¹⁾ … yₙ⁽ⁿ⁻¹⁾ ],
row k holding the (k − 1)th derivatives.
Derivatives. Each derivative comes from a computer algebra system (nerdamer, open source); the algebra runs after you start typing, in the background. Each is checked: at 15 fixed test points (0.21, −0.37, 0.57, 1.33, −0.91, 2.63, −1.73, 0.77, 4.19, −3.17, 1.61, −0.043, 3.37, −7.31, 8.93), a five-point central difference with steps h = 0.001 × max(1, |x|) and h/2 must match the formula to 10⁻⁶ × max(1, |slope|) (plus 10⁻¹¹ × |f| / h for rounding). Points where the function is not real, or where the two steps disagree, are skipped; at least 3 must agree and none may disagree.
Determinant. W is expanded by cofactors along the first row, each minor simplified first. A form is simplified by three candidates: the form itself; the algebra’s expanded form; and that expanded form expanded again after each cos(u)² is written as 1 − sin(u)². The shortest written form is kept, and each expanded form must equal the determinant at the test points to 10⁻⁹. So 2 cos(2x)² + 2 sin(2x)² becomes 2.
0 at every test point. When, at every test point where W is defined, |W| is at most 10⁻⁹ times the sum of the absolute values of the terms of W, the verdict is that the test does not show independence (for solutions of one linear differential equation, they are then dependent). Otherwise the verdict is that the functions are linearly independent. W itself is shown as simplified, 0 when the algebra reduces it to 0. The written W is read back and must equal the checked W at the test points (both real and within 10⁻⁹, or neither real).
What you can type
- 2 to 4 functions, separated by commas, in one letter (any lowercase letter except e, which is Euler’s number). A constant function such as 1 is allowed.
- Operations: + − * / and ^; brackets group; numbers and letters side by side multiply.
- Functions: sqrt, cbrt, ln (and log), log10, exp, abs, sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh and their inverses. Angles are in radians.
How answers are written
- Wronskian W: one expression in the syntax you type (^ for powers, a space for multiplication, e^(−u) possibly as 1/e^u), as MathML, with its terms in the order the algebra gives them. An answer whose text holds a decimal point or a whole number of 9 or more digits (a number the algebra rounded) is not shown.
- Linear independence: "Linearly independent: W is not 0 everywhere." or "Not shown: W is 0 at every test point, so the test does not show independence."
Assumptions
- Angles are in radians; ln and log are the natural logarithm.
- An answer that fails its check, finds no formula, or takes over 3 seconds is not shown ("No verified answer").
Worked examples by hand
y₁ = eˣ, y₂ = e⁻ˣ (Trench, section 5.1, Example 5.1.5 (a)). W = eˣ · (−e⁻ˣ) − eˣ · e⁻ˣ = −1 − 1 = −2, never 0, so the functions are linearly independent.
y₁ = x², y₂ = 1/x² (Example 5.1.5 (c)). y₁′ = 2x and y₂′ = −2/x³, so W = x² · (−2/x³) − 2x · (1/x²) = −2/x − 2/x = −4/x, not 0 for x ≠ 0.
eˣ, e²ˣ, e³ˣ (by hand). The rows are (eˣ, e²ˣ, e³ˣ), (eˣ, 2e²ˣ, 3e³ˣ), (eˣ, 4e²ˣ, 9e³ˣ). Taking e⁶ˣ out, W = e⁶ˣ · det[1, 1, 1; 1, 2, 3; 1, 4, 9] = e⁶ˣ · 2 = 2e^(6x).
Other questions people ask
What is the Wronskian?
For functions y₁, …, yₙ of x, the Wronskian W is the determinant of the n × n matrix whose first row is y₁, …, yₙ, whose second row is their first derivatives, and so on up to the (n − 1)th derivatives. For two functions, W = y₁y₂′ − y₁′y₂.
How does the Wronskian test linear independence?
If W is not 0 at some point of an interval, the functions are linearly independent on that interval. For e^x and e^(−x), W = −2, so they are independent. The converse needs care: W = 0 everywhere does not by itself prove dependence (x² and x|x| have W = 0 but are independent), but for solutions of one linear differential equation it does.
Why is the Wronskian used with differential equations?
Solutions y₁, y₂ of a second-order linear equation y″ + p(x)y′ + q(x)y = 0 form a fundamental set, so every solution is c₁y₁ + c₂y₂, exactly when their Wronskian is not 0. Abel’s formula shows that such a Wronskian is either never 0 or always 0 on an interval.
How is the answer checked?
Each derivative comes from a computer algebra system and must match a numeric difference quotient at 15 test points. The determinant is built from those checked derivatives, and each simplified form of it must equal that determinant at the same points. An answer that fails is not shown.
How does the page decide that W is 0?
W counts as 0 when, at every one of 15 test points where it is defined, its value is within 10⁻⁹ of the sum of the sizes of the terms of the determinant. The page then says that the test does not show independence. W itself shows as 0 when the algebra reduces it to 0, and otherwise as the expression, such as sin(x)² + cos(x)² − 1.
Which functions can I type?
Any functions of one letter (x, t, or another letter except e), separated by commas: polynomials, e^(kx), sin, cos, ln, sqrt and so on. A constant such as 1 is allowed. Two different letters give a message.