# 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)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)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)