Wronskian Determinants and Higher Order Linear Hom. Diff. Eqs.

Wronskian Determinants and Higher Order Linear Homogenous Differential Equations

Recall from the Wronskian Determinants and Linear Homogenous Differential Equations page that if we have a second order linear homogenous differential equation $\frac{d^2y}{dt^2} + p(t) \frac{dy}{dt} + q(t)y = 0$, then we saw that if $y = y_1(t)$ and $y = y_2(t)$ are both solutions to this differential equation, then for $C$ and $D$ as constants we had that the linear combination $y = Cy_1(t) + Dy_2(t)$ was also a solution to this differential equation.

We then went on to question whether all solutions to this differential equation are given in this form. If we had the initial conditions $y(t_0) = y_0$ and $y'(t_0) = y_0'$, then $y = Cy_1(t) + Dy_2(t)$ is the solution corresponding to this differential equation and the initial value problem if we have that:

(1)

\begin{align} \quad Cy_1(t_0) + Dy_2(t_0) = y_0 \\ \quad Cy_1'(t_0) + Dy_2'(t_0) = y_0' \end{align}

Such unique constants $C$ and $D$ exist when the Wronskian determinant $W(y_1, y_2) = \begin{vmatrix} y_1(x) & y_2(x)\\ y_1'(x) & y_2'(x) \end{vmatrix} \neq 0$.

The same sort of approach can be used if we have an $n^{\mathrm{th}}$ order linear homogenous differential equation $\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0$ and suppose that $y = y_1(t)$, $y = y_2(t)$, …, and $y = y_n(t)$ are all solutions to this differential equation. Then for $C_1$, $C_2$, …, $C_n$ as constants, we want to see whether or not all solutions to this differential equation are given in the form:

(2)

\begin{align} \quad y = C_1y_1(t) + C_2y_2(t) + ... + C_ny_n(t) \end{align}

If this is the case, when for any defined $t_0$ with the initial conditions $y(t_0) = y_0$, $y'(t_0) = y_0'$, …, $y^{(n-1)}(t_0) = y_0^{(n-1)}$ we must have that the constants $C_1$, $C_2$, …, $C_n$ all satisfy the following system of equations:

(3)

\begin{align} \quad C_1y_1(t_0) + C_2y_2(t_0) + ... + C_ny_n(t_0) = y_0 \\ \quad C_1y_1'(t_0) + C_2y_2'(t_0) + ... + C_ny_n'(t_0) = y_0' \\ \quad \quad \quad \quad \quad \quad \vdots \quad \quad \quad \quad \quad \quad \\ \quad C_1y_1^{(n-2)}(t_0) + C_2y_2^{(n-2)}(t_0) + ... + C_ny_n^{(n-2)}(t_0) = y_0^{(n-2)} \\ \quad C_1y_1^{(n-1)}(t_0) + C_2y_2^{(n-1)}(t_0) + ... + C_ny_n^{(n-1)}(t_0) = y_0^{(n-1)} \\ \end{align}

This is a system of $n$ equations in the $n$ unknown constants $C_1$, $C_2$, …, $C_n$, and so from linear algebra, we know that a unique solution for the constants exists provided that the determinant of the corresponding coefficient matrix is nonzero, that is:

(4)

\begin{align} \quad \begin{vmatrix} y_1(t_0) & y_2(t_0) & \cdots & y_{n-1}(t_0) & y_n(t_0)\\ y_1^{(1)}(t_0) & y_2^{(1)}(t_0) & \cdots & y_{n-1}^{(1)}(t_0) & y_n^{(1)}(t_0)\\ \vdots & \vdots & & \vdots & \vdots\\ y_1^{(n-2)}(t_0) & y_2^{(n-2)}(t_0) & \cdots & y_{n-1}^{(n-2)}(t_0) & y_n^{(n-2)}(t_0) \\ y_1^{(n-1)}(t_0) & y_2^{(n-1)}(t_0) & \cdots & y_{n-1}^{(n-1)}(t_0) & y_n^{(n-1)}(t_0)\\ \end{vmatrix} \neq 0 \end{align}

Note though that the determinant above is simply the Wronskian determinant of $f_1$, $f_2$, …, $f_n$ evaluated at $t_0$, that is the determinant above is $W(f_1, f_2, ..., f_n) \biggr \rvert_{t_0}$. Now provided that the Wronskian determinant of $f_1$, $f_2$, …, $f_n$ is not zero then we can solve for the constants $C_1$, $C_2$, …, $C_n$ for using Cramer's rule. For each $j = 1, 2, ..., n$, the value of the constant $C_j$ is obtained by taking the determinant of the Wronskian matrix evaluated at $t_0$ whose $j^{\mathrm{th}}$ column is replaced by $\begin{bmatrix} y_0\\ y_0'\\ \vdots\\ y_0^{(n-2)}\\ y_0^{(n-1)} \end{bmatrix}$, and then this result divided by the Wronskian Determinant $W(f_1, f_2, ..., f_n) \biggr \rvert_{t_0}$.

Once again, we want to know whether or not every solution to this $n^{\mathrm{th}}$ order linear homogenous differential equation is of the form $y = C_1y_1(t) + C_2y_2(t) + ... + C_ny_n(t)$. The following theorem guarantees us a sufficient condition for this property.

Theorem 1: Let $\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0$ is an $n^{\mathrm{th}}$ order linear homogenous differential equation where $p_1$, $p_2$, …, $p_n$ are all continuous functions on an open interval $I$. Then if $y = y_1(t)$, $y = y_2(t)$, …, $y = y_n(t)$ are solutions to this differential equation, then every solution to this $n^{\mathrm{th}}$ order linear homogenous differential equation can be expressed as a linear combination $y = C_1y_1(t) + C_2y_2(t) + ... + C_ny_n(t)$ if $W(y_1, y_2, ..., y_n) \neq 0$ for at least one point $t \in I$.

Like before, we will bring up the term, "fundamental set of solutions" as the set of solutions $y = C_1y_1(t) + C_2y_2(t) + ... + C_ny_n(t)$ when every solution to this differential equation is of this form.

Definition: The set of solutions $y = y_1(t)$, $y = y_2(t)$, …, $y = y_n(t)$ form a Fundamental Set of Solutions to the differential equation $\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0$ if $W(f_1, f_2, ..., f_n) \neq 0$.