Wronskian

In mathematics, the Wronskian can be used to determine if a set of functions are linearly independent. For a set of functions, <math>y_1(t),y_2(t),...,y_n(t)</math>, the Wronskian is defined by:

<math>

W(y_1,y_2,...,y_n) = \begin{vmatrix}

 y_1 & y_2 & \cdots & y_n \\
 y'_1 & y'_2 & \cdots & y'_n \\
 \vdots  & \vdots  & \ddots & \vdots  \\
 y^{n-1}_1 & y^{n-1}_2 & \cdots & y^{n-1}_n 

\end{vmatrix} </math>

where bars indicate the determinant of the matrix. Recall from linear algebra that, if the determinant of a matrix is nonzero, it means that the two columns of the matrix are linearly independent of each other. Hence if the Wronskian is not zero, then the functions are linearly independent. Otherwise if it is zero, then they may or may not be linearly independent. Furthermore, if over some range the Wronskian is non zero, the functions are linearly independent over that range.

Differential Equations

A n^th order linear differential equation will have n linearly independent solutions. For a second order equation of the form <math>y + p(t)y' + q(t)y = 0</math>, the Wronskian is particularly useful as when one solution, <math>y_1</math> is known, the other, linearly independent solution <math>y_2</math> can be easily found. The Wronskian will be:

<math>W(y_1, y_2) =

det \begin{bmatrix}

 y_1      & y_2      \\
 y_1' & y_2'

\end{bmatrix} = y_1 y_2' - y_2 y_1' </math>

Differentiating the Wronskian with respect to t:

<math>

W'(t) = y_1'y_2' + y_1y_2 - y_2'y_1' - y_2y_1 = y_1y_2 - y_2y_1 </math>

As both <math>y_1</math> and <math>y_2</math> solve the differential equation, eliminating the second order derivatives gives:

<math>

W'(t) = (py_1'+qy_1)y_2 - y_1(py_2'+qy_2) =-p(y_1y_2'-y_1'y_2) = -pW </math>

The Wronskian can therefore be found from <math>p(t)</math> as:

<math>

W(t) = C \exp{-\int^t p(u) \, du} </math>

This is known as Abel's identity. It leads to a useful method for solving second order differential equations. As soon as one solution, <math>y_1(t)</math>, is known, the Wronskian can be calculated using the equation above. Noting that for a second order equation, the Wronskian can be expressed as:

<math>

W(t) = = y_1 y_2' - y_2 y_1' = y_1^2 \frac{d}{dt} \left( \frac{y_2}{y_1} \right) </math>

rearranging and integrating gives an expression for solving a second order equation:

<math>

y_2(t) = y_1 \int^x \frac{W(u)}{y_1^2(u)} \, du = y_1 \int^x \frac{\exp{\left(-\int^u p(v) \, dv\right)}}{y_1^2(u)} \, du </math>

See also

• Exact differential equation
• Euler substitution
• Reduction of order