Orthogonal Latin Squares of Order 5

Problem
How can two orthogonal Latin squares of order $5$ be constructed?

Solution
The elements of the first Latin square are shown in normal type, while those of the second Latin square are shown in italic type:

$\begin{array} {|c|c|c|c|c|} \hline 1 \mathit 1 & 2 \mathit 3 & 3 \mathit 5 & 4 \mathit 2 & 5 \mathit 4 \\ \hline 3 \mathit 2 & 4 \mathit 4 & 5 \mathit 1 & 1 \mathit 3 & 2 \mathit 5 \\ \hline 5 \mathit 3 & 1 \mathit 5 & 2 \mathit 2 & 3 \mathit 4 & 4 \mathit 1 \\ \hline 2 \mathit 4 & 3 \mathit 1 & 4 \mathit 3 & 5 \mathit 5 & 1 \mathit 2 \\ \hline 4 \mathit 5 & 5 \mathit 2 & 1 \mathit 4 & 2 \mathit 1 & 3 \mathit 3 \\ \hline \end{array}$