Exponential Model

Goal: given $\frac{dy}{dx}\propto y$, $y(0)=y_0$, $y(T)=y_T$ and $T\ne0$, find $y(x)$ in terms of $y_0$, $y_T$ and $T$.

$\begin{aligned} \frac{dy}{dx}&\propto y\\ \frac{dy}{dx}&=ky\\ \int \frac1y\,dy&=\int k\,dx\\ \ln|y|&=kx+C\\ y&=\exp(kx+C)\\ y&=\exp(C)\exp(kx)\\ y(x)&=C\exp(kx)\\ \end{aligned}$

$\begin{aligned} y(x)&=C\exp(kx)\\ y(0)&=C\exp(k(0))\\ y_0&=C\exp(0)\\ y_0&=C(1)\\ y_0&=C\\ \end{aligned}$

$\begin{aligned} y(x)&=C\exp(kx)\\ y(x)&=y_0\exp(kx)\\ y(T)&=y_0\exp(kT)\\ \frac{y_T}{y_0}&=\exp(kT)\\ \ln\left(\frac{y_T}{y_0}\right)&=kT\\ \frac1T\ln\left(\frac{y_T}{y_0}\right)&=k\\ \end{aligned}$

$\begin{aligned} y(x)&=C\exp(kx)\\ y(x)&=y_0\exp\left(\frac1T\ln\left(\frac{y_T}{y_0}\right)x\right)\\ y(x)&=y_0\left(\exp\left(\ln\left(\frac{y_T}{y_0}\right)\right)\right)^{\frac1Tx}\\ \end{aligned}$

$\therefore\boxed{y(x)=y_0\left(\frac{y_T}{y_0}\right)^{\frac xT}}$

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$