Physics and Differential Equations

class: center, middle, inverse, title-slide

.title[
# Physics and Differential Equations
]
.subtitle[
## Session 04
]
.author[
### Alejandro Ucan
]
.date[
### 2022-10-15
]

---

# Goals:

* Broaden the type of Differential Equations. <br/><br/>
 * State physical systems via differential equations. <br/><br/>
 * Look for solutions.
 
---
# A key example.

> Imagine that we have a contaminated slice of bread with some bacteria. After some observations, we can assure that the growing rate of the population is 3 times each second. <br/><br/>
If `$P(t)$` denotes the measure of the bacteria population (by thousands), then the __growing rate__ corresponds to `$$\frac{dP}{dt}.$$` <br/><br/>
Following our observations, we can state that `$$\frac{dP}{dt}=3P.$$`

---
## Information of our problem.

> Recall that (from Differential Calculus) there is one function whose derivative its almost the same as the function (up to a constant). If `$f(t)=e^{3t},$` then `$$\frac{df}{dt}=3e^{3t}=3f(t).$$` <br/><br/> But this also works for the function `$f_1(t)=2e^{3t},$` `$f_1(t)=10e^{3t},$` etc.

---
## A picture of this.

---
# A physical model.

> Assume that we have an object of mass `$m$` attached at the end of a spring, and the other end of the spring is fixed. <br/><br/> Hooke's law says that:<br/> If a spring is stretched (compressed) `$x$` units from its natural length, then it exerts a force that is proportional to `$x:$` `$$\mbox{restoring force}=-kx.$$`
By Newton's second law (ignoring all other external forces), says that _force is equal mass times acceleration_ `$$m\frac{d^2x}{dt^2}=-kx.$$`

---
## The mass-spring model

> Previous differential equations is a model of the physical phenomena. This differential equation doesn't look like solvable. To find its solution we need more refinated techniques for differential equations theory. <br/><br/>
We can picture what should happend.

---
## A picture of the phenomenon

---
# A less-ideal model

> Let's assume that in our last model we consider the presence of a new external force, for example the existence of resistance. <br/><br/> This could be think as that our spring-mass is inmersed in a viscous medium. <br/><br/> In mechanics, the force induced by the viscous medium acts proportionally to the instant velocity. Our model transforms to `$$m\frac{d^2x}{dt^2}=-kx-\beta \frac{dx}{dt}.$$`

---
## Lets see some pictures.