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.

Details and Options

  • The Wronskian determinant is defined as Det[Table[D[yi,{x,j}],{i,m},{j,0,m-1}]].
  • If the functions y1,y2,… are linearly dependent, the Wronskian vanishes everywhere.

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:

Scope  (9)

Functions  (6)

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:

Differential Equations  (3)

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:

Wolfram Research (2008), Wronskian, Wolfram Language function, https://reference.wolfram.com/language/ref/Wronskian.html.

Text

Wolfram Research (2008), Wronskian, Wolfram Language function, https://reference.wolfram.com/language/ref/Wronskian.html.

CMS

Wolfram Language. 2008. "Wronskian." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Wronskian.html.

APA

Wolfram Language. (2008). Wronskian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Wronskian.html

BibTeX

@misc{reference.wolfram_2024_wronskian, author="Wolfram Research", title="{Wronskian}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Wronskian.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_wronskian, organization={Wolfram Research}, title={Wronskian}, year={2008}, url={https://reference.wolfram.com/language/ref/Wronskian.html}, note=[Accessed: 21-December-2024 ]}