Exponential Growth Equation/Special Case

Theorem
All solutions of the differential equation $y' = y$ take the form $y = C e^x$.

Proof
Let $\map f x = C e^x$.

Then by Derivative of Exponential Function:


 * $\map {f'} x = \map f x$

From Exponential of Zero:


 * $\map f 0 = C$

Hence $C e^x$ is a solution of $y' = y$.

Now suppose that a function $f$ satisfies $\map {f'} x = \map f x$.

Consider $\map h x = \map f x e^{-x}$.

By the Product Rule for Derivatives:

From Zero Derivative implies Constant Function, $h$ must be a constant function.

Therefore, $\map h x = \map h 0 = \map f 0$.

Recalling the definition of $h$, it follows that:


 * $\map f x = \map f 0 e^x$