Decay Equation

Theorem
The first order ordinary differential equation:
 * $$\frac{\mathrm{d}{y}}{\mathrm{d}{x}} = k \left({y_a - y}\right)$$

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:
 * $$y = y_a + \left({y_0 - y_a}\right) e^{-k x}$$

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:
 * $$y = y_a + \left({y_0 - y_a}\right) e^{-k x}$$

Also see

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