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: ...

I wanted to prove:

  if .

using first principle, but without any success. Can anyone help me, and show me the detail steps?

This equation is an inhomogeneous version of the equation in this post.

How to solve this equation:

This question is inspired by this post.

Compute:

where a is any complex number, while N is a positive integer.

I know that the solution can be written as: ...

It is a forward of question in this post.

I used the following code to solve a differential equation:

I would like to make a function out of the output. But I have no idea how to do that. Thank you.

This question is a forward from this post.

I am using the equation below as equation of circle in polar coordinate.

Here, R is the distance between the origin and the centre of the circle, A is the radius ...

This question is a forward of this post.

What is the recurrence relation of this sequence? And it's solution?

I stumbled across this post, and I can't figure it out myself.

Evaluate:

I am not very sure how to handle this kind of problem. Thank you.