This is a short how to about how to sum sin(n q) and cos(n q).

Given that A is a matrix, I would like to ask how to show:

Thank you very much.

I would like to know how to solve the following differential equation (you may refer to this post for the original question):

Thank you very much.

This question is a mirror of the question Sum of subsets which I found interesting. The problem is like this:

Let , how many subsets with k elements that are having the sum of value r?

For instance, if n = 6, k = 3, r = 10:

The following sets: ...

This paper is inspired by this post. It asked how to count the number of solutions of the equation with non-negative integral solution:

x_1 + ... + x_k = n

without using the standard method of line and dot counting. I transform the problem into a partial difference equation, ...

I tried to find the n-th derivative of the function sec x + tan x in a specific form, and a partial difference equation arises from it. An attemp to solve the equation is stated in another article: Reducing a partial difference equation into a partial differential equation and solving for the generating function using method of characteristics

I am going to solve a partial difference equation, firstly to transform it into a partial differential equation by using generating function method. Then, the partial differential equation can be solved by using method of characteristics.

In this paper, I will derive an explicit formula for the Euler zigzag numbers (Up/down numbers). Euler zigzag number is the number of alternating permutation in a set. Therefore the explicit formula of Euler numbers(Secant numbers) and Bernoulli numbers are found as well. The formula involves two finite sum.

In this short how-to, I will show how to solve 2nd-order differential equation of the form:

y''(t) = f(y)

I will give an example at the end.

In this how-to, I am going to solve a recurrence differential equation using Binomial Expansion Theorem for non-commutative elements.