Solve the separable differential equation $\frac{d y}{d x}=-0.8 y$, and find the particular solution satisfying the initial condition $y(0)=-7$. \[ y(x)= \]

Step 1 ：This is a first order linear differential equation. The general solution to this type of equation is given by \(y(x) = Ce^{kx}\), where \(C\) is a constant and \(k\) is the coefficient of \(y\) in the differential equation. In this case, \(k = -0.8\).

Step 2 ：To find the particular solution satisfying the initial condition \(y(0) = -7\), we need to substitute \(x = 0\) into the general solution and solve for \(C\).

Step 3 ：Substituting \(x = 0\) into the general solution gives \(C = -7\).

Step 4 ：Substituting \(C = -7\) into the general solution gives the particular solution \(y(x) = -7e^{-0.8x}\).

Step 5 ：\(\boxed{y(x) = -7e^{-0.8x}}\) is the particular solution to the differential equation \(\frac{d y}{d x}=-0.8 y\) that satisfies the initial condition \(y(0)=-7\).