## Differential equation

Let be a twice differentiable function such that

If , find an explicit formula of .

**Solution**

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.