The Wronskian of a Linear Homogeneous nth Order ODE

# The Wronskian of a Linear Homogeneous nth Order ODE

Recall from the Fundamental Sets and Matrices of a Linear Homogeneous nth Order ODE page that if we have a linear homogeneous $n^{\mathrm{th}}$ order ODE $y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0y = 0$ then a linearly independent set of solutions $\{ \psi_1, \psi_2, ..., \psi_n \}$ to this ODE is called a fundamental set of solutions, and the corresponding fundamental matrix is given by:

(1)
\begin{align} \quad \Psi(t) = \begin{bmatrix} \psi_1(t) & \psi_2(t) & \cdots & \psi_n(t) \\ \psi_1'(t) & \psi_2'(t) & \cdots & \psi_n'(t) \\ \vdots & \vdots & \ddots & \vdots \\ \psi_1^{(n-1)}(t) & \psi_2^{(n-1)}(t) & \cdots & \psi_n^{(n-1)}(t) \end{bmatrix} \end{align}

We give a special name to the determinant of such matrix.

 Definition: The Wronskian of a linear homogeneous $n^{\mathrm{th}}$ order ODE $y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0y = 0$ with Fundamental matrix $\Psi$ is defined to be $W(\psi_1, \psi_2, ..., \psi_n)(t) = \det \Psi (t)$.

Recall that for linear homogeneous systems of first order ODEs $\mathbf{x}' = A(t) \mathbf{x}$, we have that the determinant of a fundamental matrix $\Phi(t)$ is given by:

(2)
\begin{align} \quad \det \Phi(t) = \det \Phi (\tau) \cdot \mathrm{exp} \left ( \int_{\tau}^{t} \mathrm{tr}(A(s)) \: ds \right ) \end{align}

For a linear homogeneous $n^{\mathrm{th}}$ order ODE, we have that the matrix $A(t)$ is the companion matrix for this system, and $\mathrm{tr}(A(t)) = -a_{n-1}(t)$. Therefore, the Wronskian for the fundamental matrix $\Psi$ is given by:

(3)
\begin{align} \quad W(\psi_1, \psi_2, ..., \psi_n)(t) = W(\psi_1, \psi_2, ..., \psi_n)(\tau) \cdot \mathrm{exp} \int_{\tau}^{t} -a_{n-1}(s) \: ds \end{align}