n-th order linear differential equations\(\frac{d^ny}{dx^n}\)

order=n, degree=1

Homogeneous, non-homogeneous linear differential equation

https://mathemerize.com/what-is-homogeneous-function-definition-and-example/

Linear combination solutions is a solution.

This is for the homogeneous linear part of nth order linear equation.

Linear independence of solutions suffices definition of linear independent functions.

Wronskian test for dependence

Wronskian determinant

https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx

We have a a linear homogeneous differential equation, just to test if the solutions are independent.

General solution of an n-th order linear differential equation

Linear combination solutions from the homogeneous linear part is called the complementary function. There is also a particular solution \(y_p(x)\) is called a particular integral. n-th order linear differential equation is complicated, we will develop more details in the rest write-ups.