Prof. Alejandro Ucan-Puc
Definition: A model is a simplified (mathematical) representation of a real problem.
Hyphotesis: The infection rate of a virus in cells is proportional to the number of infected cells.
Variables and parameters:
Equation: Remember that the growth rate is a rate of change: $$\frac{dP}{dt}=kP.$$
Hyphotesis 1: If the population of infected cells is small then the growth rate is proportional to the elements of the population. But if the population is too large (to be supported), the population will begin to decrease.
Variables and parameters:
Equation: $$ \frac{dP}{dt}=kP(1-\frac{P}{N}).$$
Definition: An ordinary differential equation is an equation whose unknown is a function and involves derivatives (of the unknown) and the independent variable. $$f(y^{(n)},y^{(n-1)},\cdots, y',y,x)=0$$
Definition: We say that $f$ is a function defined on a certain interval $I,$ with continuous derivatives in $I$ (as many derivatives as the order of the equation) is a solution of a differential equation if when we substitute the function we obtain an identity.
Example: The function $f(x)=e^x$ defined in $\mathbb{R},$ is a solution of the ODE $$y'-y=0.$$ Let's see, $f'(x)=e^x$ and if $y=f(x),$ then $$y'-y=e^x-e^x=0.$$ Is it true if we substitute $y=4e^x$ in the ODE $y'-y=0$?
Definition: The graph of a function solution to an ODE is known as a solution curve. Note that the solution curve may differ from the graph of the function given that the solution curve depends on the interval of definition of the solution.
Example: Consider the equation $xy'+y=0.$ The function $y=f(x)=1/x$ is a solution, but remember that $f(x)$ is not differentiable at $x=0,$ so we must define the interval we will take, if $(-\infty,0)$ or $(0,\infty).$
Definition: Sometimes the ODE can be satisfied by a relationship $G(x,y)$ (equation involving $x$ and $y), but always with the premise that there is a function that satisfies this relationship that will behave as the solution of the ODE.
Example: The relationship $x^2+y^2=k^2$ is a solution of the equation $yy'+x=0.$ And the associated functions are $f(x)=\sqrt{k^2-x^2}$ and $f(x)=-\sqrt{k^2-x^2}$ in their respective intervals of definition.
Hyphotesis: In these problems, the central hypothesis is that the rate of change of a certain quantity is in proportion to its population or quantitie at any moment.
The ODE that models these problems is $$\frac{dP}{dt}=kP.$$
Hyphotesis: In these problems, the central hypothesis is that the rate of change of a certain quantity is in proportion to the difference between the quantity and the quantity of the medium in which it is found.
The ODE that models these problems is $$\frac{dP}{dt}=k(P-P_m).$$
Temperature hypotheses: The rate of change of the temperature that a body experiences is proportional to the difference between the temperature of the body and the temperature of the medium in which it is found.
Example: A motor's temperature is placed in a room with a temperature of 20°C. When the motor is turned on, the temperature of the motor is 30°C. The motor's temperature increases at a rate of 2°C per minute. What will be the temperature of the motor after 10 minutes?
Mixtures Hypotheses: In a tank with $N$ liters of water mixed with an agent (salt, sugar, etc.). Consider the following:
$$\frac{d S}{dt}=ak-b\frac{S}{N}.$$
Definition: The qualitative analysis of an ODE consists of studying the behavior of the solutions of the ODE without solving it.
Definition: The slope field of the ODE $y'=f(x,y)$ is constructed with a representation of a line with slope $f(x,y)$ at each point $(x,y).$
Study the slope field of the following ODEs:
The solution curves are the curves that follow the direction of the slopes of the slope field.
We can classify these curves by:
Study the slope field of the ODE $y'=4y(1-y).$
Fundamental definitions, mathematical modeling principles, rate of growth/decay hypotheses, qualitative analysis, and slope fields.
Next Session: Non-homogeneous 1st Order Linear ODEs