Determinant

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Wronskian Determinants of Two Functions

We are going to look more into second order linear homogenous differential equations, but before we do, we need to first learn about a type of determinant known as a Wronskian Determinant which we define below.

Definition: Let and be two differentiable functions. Then the Wronskian Determinant of and is the determinant .

Sometimes the term "Wronskian" by itself is used to mean the same thing as "Wronskian Determinant". Furthermore, sometimes we can just write "", or "" instead of to represent the Wronskian of and .

Let's look at some examples of computing the Wronskian determinant of two differentiable functions.

Example 1

Determine the Wronskian of the functions and . For what values of is the Wronskian equal to zero?

We note that and are both differentiable functions and that and . Therefore the Wronskian of and is:

Therefore the Wronskian of and is equal to zero for all .

Example 2

Determine the Wronskian of the functions and . For what values of is the Wronskian equal to zero?

We note that and are both differentiable functions and that and . Therefore the Wronskian of and is:

Note that for all so the Wronskian of and is zero nowhere.

Wronskian Determinants and Linear Homogenous Differential Equations

Recall that if we have the second order linear homogenous differential equation and and are solutions to this differential equation, then for constants and , the linear combination is also a solution to this differential equation. One question to ask if whether or not all of the solutions to this differential equation are in this form as we do not want to miss any other potential solutions.

Suppose that we are given an initial value problem to a second order linear homogenous differential equation in this form alongside the initial conditions and . Applying the first initial condition to our solution and we have that:

Now the derivative of is . Applying the second initial condition and we have that:

We need the unknown constants and to satisfy both of these equations, that is, we want to solve the system for the constants and . We note that we have a system of two equations with two unknowns and thus, a unique solution exists if the determinant of the augmented matrix for this system is nonzero, that is:

Notice though that this determinant is simple the Wronskian of the functions and evaluated at . If , then we can apply Cramer's Rule from linear algebra to find the values of the constants and . We have that:

If this determinant equals zero, then either no solutions for and exist, or infinitely many solutions for the values of and may exist. The following theorem summarizes what we have just found.

Theorem 1: Let be a second order linear homogenous differential equation where and are continuous functions on an open interval such that and with the initial conditions and . If and are solutions to this differential equation then there exists constants and for which is a solution to the initial value problem if and only if the Wronskian at is nonzero, that is .