Final answer:
To solve the separable differential equation dy/dx = -0.3y and find the particular solution satisfying the initial condition y(0) = 1, we can separate the variables and integrate both sides. The solution to the initial value problem is y(x) = e^(-0.3x).
Step-by-step explanation:
To solve the separable differential equation dy/dx = -0.3y, we can separate the variables and integrate both sides. Starting with dy/dx = -0.3y, we can rearrange the equation as dy/y = -0.3dx. Integrating both sides gives us ln|y| = -0.3x + C, where C is a constant. To find the particular solution satisfying the initial condition y(0) = 1, we substitute x = 0 and y = 1 into the equation and solve for C. ln|1| = -0.3(0) + C, so C = 0. The solution to the initial value problem is y(x) = e^(-0.3x).