Decay Equation

Theorem
The first order ordinary differential equation:
 * $\dfrac {\d y} {\d x} = k \paren {y_a - y}$

where $k \in \R: k > 0$

has the general solution:
 * $y = y_a + C e^{-k x}$

where $C$ is an arbitrary constant.

If $y = y_0$ at $x = 0$, then:
 * $y = y_a + \paren {y_0 - y_a} e^{-k x}$

This differential equation is known as the decay equation.

Proof
This is our general solution.

Suppose we have the initial condition:
 * $y = y_0$ when $x = 0$

Then:
 * $y_0 = y_a + C e^{-k \cdot 0} = y_a + C$

and so:
 * $C = y_0 - y_a$

Hence the solution:
 * $y = y_a + \paren {y_0 - y_a} e^{-k x}$

Also see

 * Newton's Law of Cooling
 * First-Order Reaction