Continuing with the topic of the complex numbers, the professor talks about two kinds of solutions for a specific type of differential equation, y’+ky=kcosωt.
With further teachings about the complex numbers, the lecture keeps elaborating on solving certain differential equations by using complex numbers. After some of my own research, I found the history of complex numbers quite interesting. Continue reading
In the following lecture, the professor reintroduced the idea of complex numbers and its application in differential equations, which I found very intriguing.
Complex number, for those who are not familiar to the concept, is a number that does not exist in any visible or “real” world. It is not used in daily life, because it does not really “exists”. Many of the beginner level math students are taught that there can be no square root for any negative numbers, nor can there be any negative squared numbers. When they are asked what is the square root of -1, they would say: “There is none”. Continue reading
During this week’s lecture, the professor talks about the idea of autonomous differential equations, which is essentially the differential equations without independent variables, namely x. The general look of such equation should be dy/dx=f(y). I was confused at first: how can equations regarding x be without it? Then I realized it is a derivative rather than a equation, so its integral would naturally be with x, just like the integral dy/dx=2 would be y=2x+c. (Click here for more information about autonomous differential equation) Continue reading
The lecture this week involved the substitution of variables as the means of changing something insolvable into a function that is solvable. There are very specific methods to solve first order differential equations. They are like formulas; every variable or part of the function must be arranged in order to make the function solvable. When a function is seemingly unsolvable in its form, the substitution of variables, in most cases the substitution of the variable “Y” or a certain function of X, is the most common way to make the miracle happen. (For more information on substitution in differential equations click here.) Continue reading
This week the professor finally reached the content that actually relates to the course’s name. He introduced first-order linear equations. It seemed such an abstract idea to me after I finished the lecture, but with further research and study, I found that such an equation is highly applicable in the real world. For more detail about first order linear equations click here
this picture shows the general solution formula for all first order linear equations (picture from http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx, which also explains the process very well)
After the introduction of Euler’s method, I did some research on numerical method and numerical analysis, and found it very much intriguing. By definition, numerical analysis is the study of numbers, the approximation techniques for solving mathematical problems. Unlike other topics, numerical methods have an extremely wide range for real life application. Continue reading
In today’s lecture the professor reintroduced “the Euler’s method”, which is a way using a start point and the first order differential equation of the function to simulate the real curve of the function. Some may wonder: what is the use of simulation of a curve when the curve is known, Continue reading
In the lecture, the “existence and the uniqueness theorem” of differential equations was mentioned. It states that on any point on the coordinate system (X0, Y0) of the differential equation y’=f(x, y) has one and only one solution through the point, meaning for any point on the direction field there is only one slope for the integral line (http://www.sosmath.com/diffeq/first/existence/existence.html). Continue reading
The first lecture I listened is about the geometric view of first-degree differential equations and the manual way of creating a direction field.
Function y’=f(x, y) can be shown on the coordinate system in the form of direction field, and one solution of y’=f(x, y), y1(x), shows as one integral curve on the coordinate system. Just as y’=f(x, y) relates to y1(x) (y’ is the derivative of y1(x) and y1(x) is one of many possible integral of y’), the integral curve correspond to y1(x) should relate to the direction field correspond to the differential equation y’=f(x, y); to be specific, the slope of every possible point on the function y1(x) should be the same as the slope shown on that exact point on the direction field. For example, one of the integral of the function y’=-x/y can be y=(4-x2)1/2, shown as a semicircle on the system, and for every point, (0, 2) for example, the slope of that point (which is 0) should be just the same as shown on that point of the direction field of y’=-x/y (meaning there should be a horizontal line on coordinate (0, 2) on the direction field to correspond the semicircle formed by y=(4-x2)1/2. The analytical view of the relationship between y’ and y1 should be just same as the geometric view of that between direction field and integral curve.
Furthermore, the manual way of creating a direction field is also mentioned in the lecture. The key difference between the computer way and human way is the order: the computer picks thousand or tens of thousands of points first, then calculates the slope at every single point and show them on the field. Human, however cannot possibly tolerate such massive amount of calculations. The Manual way start with picking slope, hence the value of y’, then calculate the isocline (a function) where all of its point should have the same slope on the direction field. For example, again y’=-x/y, and we pick the slope y’ of 2, then there goes a function of 2=-x/y and hence y=-x/2, and we draw the line on the coordinate system, then every point on that line should have a slope of 2, which is an extremely efficient way of creating a direction field.