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 $W(f_1, f_2, ..., f_n) \neq 0$.
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