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## Differential equation

Let be a twice differentiable function such that

If , find an explicit formula of .

Solution

We have successively

## Polylogarithm series

Let denote the polylogarithm. Prove that

Solution

Since we have successively

## Convex function

Let be a convex function on a convex domain and a convex non-decreasing function on . Prove that the composition of is convex on .

Solution

We want to prove that for it holds that

We have:

## Polynomial equation

Let denote the golden ratio. Solve the equation

Solution

First of all we note that

We easily note that is one root of the equation, hence using Horner we get that

Hence is a double root and the other root is .

## Similarity implies equality?

Let be similar to . Does hold?

Solution

No! Take then . The matrices are similar but not equal.