Separable ODE

Separable ODEs are the simplest non-trivial differential equation: when you can shuffle all the $y$'s to one side and all the $x$'s to the other, integrating both sides solves the problem.

Solve the initial value problem

$\frac{dy}{dx}=xy,y(0)=1.$

Give the explicit solution $y(x)$ and verify that it satisfies both the ODE and the initial condition.