This problem will take you through the process of solving the initial value problem for the differential equation \[ e^{x} y^{\prime}+e^{x} y=0, \quad y(0)=y_{0} \]. What does the differential equation imply about the behavior of the solution?

A. The solution is a constant function.
B. The solution is an exponential function.
C. The solution is a linear function.
D. The solution is a quadratic function.