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 3rd derivative of some kind. It does not mean that this ODE is of some third-level importance.
Here is what an nth -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 
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.
- 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.
*Note that all the equations are generated through http://www.hostmath.com