How to solve this differential equation?

How would you solve the differential equation

\(B'(x) = e^x B(x)\)

with the initial condition \(B(0) = 1\)? I know what the answer is supposed to be, but I don’t know how to directly solve it.

In case you’re wondering, \(B(x)\) is the exponential generating function for the Bell numbers, which count set partitions. The differential equation in question arises from noting that

\(\displaystyle B_{n+1} = \sum_{k=0}^n \binom n k B_{n-k}\)

(to make a partition of \(\{1, \dots, n+1\}\), you can put anywhere from \(k = 0\) to \(n\) elements in the same set with \(n+1\); there are \(\binom n k\) ways to choose the \(k\) elements to include, and \(B_{n-k}\) ways to partition the rest).