Differential Equation Proof

Show that y_c(x)=Ce^{-\int P(x)} dx is a general solution of dy/dx + P(x)y=0

I began by subtracting P(x)y from each side of the equation which produces the separable differential equation dy/dx = -P(x)y . After separating the equation, I integrate both sides of it with respect to the appropriate variables. This will look like \int(1/y) dy = \int -P(x) dx . After the integration process I will obtain the formula ln|y| = -\int P(x) + C . In order to solve for y as a function of x , I exponentiate each side of the equation to obtain the formula y = e^{-\int P(x) dx + C} . Then, using the additive power rule of exponents I can write the right hand side of the formula as the product e^{-\int P(x) dx}e^{C} . Since e^C is a constant itself, I can write that as just C and move it in front of the exponent so that my final function y(x) looks like the y_{c} function above.