Decay Equation
 | This page has been identified as a candidate for refactoring of intermediate complexity. In particular: separate out into definition and proof Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |
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
| \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds -k \paren {y - y_a}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \frac {\d y} {y - y_a}\) | \(=\) | \(\ds -\int k \rd x\) | Solution to Separable Differential Equation | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \ln {y - y_a}\) | \(=\) | \(\ds -k x + C_1\) | 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:
- $y = y_a + \paren {y_0 - y_a} e^{-k x}$
$\blacksquare$