Decay Equation

Theorem

$\dfrac {\d y} {\d x} = k \paren {y_a - y}$

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

 $\ds \frac {\d y} {\d x}$ $=$ $\ds -k \paren {y - y_a}$ $\ds \leadsto \ \$ $\ds \int \frac {\d y} {y - y_a}$ $=$ $\ds -\int k \rd x$ Separation of Variables $\ds \leadsto \ \$ $\ds \map \ln {y - y_a}$ $=$ $\ds -k x + C_1$ Primitive of Reciprocal and Derivatives of Function of $a x + b$ $\ds \leadsto \ \$ $\ds y - y_a$ $=$ $\ds e^{-k x + C_1}$ $\ds \leadsto \ \$ $\ds y$ $=$ $\ds y_a + C e^{-k x}$ where we put $C = e^{C_1}$

This is our general solution.

$\Box$

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:

$y = y_a + \paren {y_0 - y_a} e^{-k x}$

$\blacksquare$