What is the most pythonic way to solve a differential equation using the Euler method?
Using a for loop is not an unpythonic way at all. Instead, an Euler method could be implemented with a recursive function, but it not necessary and less optimized in Python.

However, methods for vectorizing recursive sequences are discussed on the numpy-discussion mailing-list. I encourage you to use structures like for loops in such situations. You will minimize the risk of errors by writing simple and concise codes.