# 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.