It may be convenient to use the following formula when modelling differential equations related to proportions:

$\frac{dy}{dt}=kM$

Where:

1. $\frac{dy}{dt}$ is the rate of change of $y$

2. $k$ is a constant

3. $M$ is the equation that models the problem

**There are many applications to first-order differential equations. Some situations that can give rise to first order differential equations are:**

• Radioactive Decay

• Population Dynamics (growth or decline)

__Exponential Model:__

$\frac{dP}{dt}=KP$

$P=Ce^{Kt}$

__Logistic Model:__

$\frac{dP}{dt}=KP(1-\frac{P}{M})$

$P= \frac{M}{1+Ce^{-kt}}$ $C= \frac{M-P_0}{P_0}$

• Newton's Law of Cooling

$\frac{dT}{dt}=-K(T-T_a)$

$|T-T_a |=Ce^{-kt}$

If $T \geq T_a$ (the object is more hot), then $T=Ce^{-kt}+T_a$

If $T$ < $T_a$ (the object is cooler), then $T=T_a-Ce^{-kt}$