Chapter 1.3 First order difference equation
In this chapter, the writer mainly introduces three different ways of solving the first order difference equation and although this chapter was constructed under the context of first order equation, the methods can be generalized to high order difference equation.
Method1: substitution, this method is plain and simple to conduct, however, it might be cumbersome to calculate in reality. But the method is theoretically available in every situation where we can calculate every item.
Method2: combination, namely derive the general solution of homogenous equation and particular solution of non-homogenous equation and combine them to form the final solution, which is a common method in the difference equation. Firstly, we need to simplify the equation into homogenous equation, this step is simple since you only need to discover and remove the constant items. And then, use some methods which will be introduced in further tutorial, we have a general solution to the homogenous equation. And we recover the equation into former non-homogenous state, and we need to commit a guess. This step is much tricker than the general solution, and it might take much more work and time. To simplify the question, we now introduce two common ways to guess the particular solution. Firstly, try to substitute the state with constant k, and you may straightly derive the value of k, which is our particular solution. And if that didn't work out, you can try to substitute it with t*k, and try to solve the equation. Above is just two common and simple ways to derive the particular solution in the first order difference equation, but for further problems, we need to apply different methods.
And here is a reminder we should keep in mind that the way we choose to derive the solution should always be accustomed to the question itself, and we only need to derive one particular solution. To have a deeper understanding of why it works, we need some knowledge in linear algebra, we imagine the polynomial to construct a linear space and we choose each power function to be the base, and if we need to handle k order equation, then we can construct k+1 dimension space, and k dimension construct the general solution and 1 dimension construct the particular space.
Method3: graphs, namely cobwebbing methods.
We draw the graph of y=x and y=f(x), and keep locate the point and we can have insights into whether the value diverges or converges.