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: $$\frac{dP}{dt}=kP.$$
Hypothesis: When the population is small, it grows proportionally to its size. When it approaches a maximum sustainable population, the growth rate decreases.
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 that function 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 $y'=4y(1-y).$
When an ODE can be written as $$\frac{dy}{dx}=g(x)h(y),$$ collect the $y$ terms with $dy$ and the $x$ terms with $dx:
$$\frac{1}{h(y)}\,dy=g(x)\,dx.$$
Solve $\frac{dP}{dt}=kP$, with $P(0)=P_0$.
For $$\frac{dP}{dt}=kP\left(1-\frac{P}{N}\right),$$ the variables separate as
$$\frac{dP}{P(1-P/N)}=k\,dt.$$
After integrating with partial fractions, applying $P(0)=P_0$, and solving for $P$:
$$P(t)=\frac{N}{1+\left(\frac{N-P_0}{P_0}\right)e^{-kt}}.$$
Fundamental definitions, mathematical modeling principles, rate of growth/decay hypotheses, qualitative analysis, and slope fields.
Next Session: Non-homogeneous 1st Order Linear ODEs