Built-in Mathematica Symbol

Wronskian

Wronskian[{y₁, y₂, ...}, x]
gives the Wronskian determinant for the functions y₁, y₂, ... depending on x.

Wronskian[eqn, y, x]
gives the Wronskian determinant for the basis of the solutions of the linear differential equation eqn with dependent variable y and independent variable x.

Wronskian[eqns, {y₁, y₂, ...}, x]
gives the Wronskian determinant for the system of linear differential equations eqns.

MORE INFORMATION

The Wronskian determinant is defined as Det[Table[D[y_i, {x, j}], {i, m}, {j, 0, m-1}]].

Linear independence of the functions y₁, y₂, ... is equivalent to the vanishing of the Wronskian.

EXAMPLES

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Basic Examples (3)

These functions are linearly independent:

These functions are dependent:

The Wronskian for a linear equation:

Except for a constant, the result is the same as for the explicit solution:

These functions are linearly independent:

These functions are dependent:

The Wronskian for a linear equation:

In[1]:=

Out[1]=

Except for a constant, the result is the same as for the explicit solution:

Scope (9)

Polynomials:

The last element can be expressed as a linear combination of the previous ones:

Rational functions:

Exponentials and exponential polynomials:

Trigonometric functions:

Special polynomials:

Other special functions:

Constant coefficient linear equation:

The Wronskian for a differential equation is usually simpler than for its solution:

Polynomial coefficient linear equation:

The corresponding Wronskian from the general solution:

Special function coefficients:

Applications (2)

Variation of parameters formula for forced second-order differential equations:

Verify that the components of the general solution for an ODE are linearly independent:

Properties & Relations (5)

Wronskian is equivalent to a determinant:

Wronskian detects linear dependence:

Casoratian performs linear dependence for sequences of a discrete argument:

Use Orthogonalize to generate a set of linearly independent functions:

Express a function in terms of the basis:

The last component is linearly dependent on the previous ones:

Use Reduce to express polynomials and rational functions in terms of each other:

Neat Examples (1)

The differential equation for Kelvin functions:

Compare to the general solution: