After learning about the first-order ODEs (ordinary differential equations) and how to solve them, it’s time to move onto higher orders. Note that in calling an ODE “third-order”, it simply means that this ODE contains a term with a 3^{rd} derivative of some kind. It does not mean that this ODE is of some third-level importance.

Here is what an *n ^{th}* -order homogeneous ODE looks like:

Now, this ODE should have n linearly independent solutions. And this is where linear algebra comes into the play. Linear independence means that no one can be written as a linear combination of others. Linear Algebra tells us that when a determinant of a square matrix is zero, then the vectors formed by each column (or row, since you can transpose them) are linearly independent. For example, consider this 2×2 matrix:

The determinant of this 2×2 matrix is -29, which is not zero. This tells us that the vectors (2,5) and (7,3) are not multiples of each other.

Wronskian is nothing but a fancy name of a special type of determinant. For example, the Wronskian of two functions, f and g, looks like this:

Like the numerical example above, the Wronskian tells us something about linear independence: whether if the function f and g are linearly independent of each other. When the Wronskian is zero, the functions are linearly dependent. The proof of this is simple. If f is indeed some constant multiple of g, then just substitute f with mg, and then the Wronskian of f and g becomes: mgg’ – mg’g = 0 ^{[2]}

Here is another proof that I came up with, but so far only applicable to a 2×2 Wronskian:

In general, a Wronskian looks like this:

where it determines the linear independence 0f n functions.

Back to the first equation. Now that homogeneous equation has n linearly independent solutions. And the Wronskian is simply a tool to verify if the solutions are correct.

The solving process of such a higher-order ODE will be discussed in the next blog post.

Works Cited:

- https://en.wikipedia.org/wiki/Wronskian
- Edwards, C. Henry, David E. Penney, and David Calvis. Elementary Differential Equations with Boundary Value Problems. Upper Saddle River, NJ: Pearson/Prentice Hall, 2008. Print.

Images Used:

- https://upload.wikimedia.org/wikipedia/commons/thumb/3/36/Jozef_Maria_Ho%C3%ABn%C3%A9-Wronski–Laurent-Charles_Mar%C3%A9chal_mg_9487.jpg/480px-Jozef_Maria_Ho%C3%ABn%C3%A9-Wronski–Laurent-Charles_Mar%C3%A9chal_mg_9487.jpg
- https://wikimedia.org/api/rest_v1/media/math/render/svg/cce1fcc4185b95fcdde1a6db7ededc8a9d8c6281

*Note that all the equations are generated through http://www.hostmath.com

Susan WaterhouseNice to see the Linear Algebra at work here. What sort of intuition or information do you have about when to expect the entries in the columns of the Wronskian matrix to be linearly independent… and what does this do for you… why do we want linearly independent vectors (functions) in this case? At some point we should have you come to Linear Algebra to talk about connections between that material and differential equations. This can be later in the year when you have learned even more connections between the two…