BUILT-IN MATHEMATICA SYMBOL

Wronskian

gives the Wronskian determinant for the functions

depending on x.

Wronskian

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

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 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:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

These functions are dependent:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

The Wronskian for a linear equation:

In[1]:=

Out[1]=

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

In[2]:=

Out[2]=

In[3]:=

Out[3]=

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: