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.)
There is a specific method used to solve first order differential equations called
“homogeneous first order differential equations (homogeneous ODEs)”, which in its standard form y’ equals some function of (y/x). In this case, by changing the homogeneous ODEs into their standard forms, a new variable “z” is created to substitute the function y/x; from that step forward, all calculations will regard the variable “z”. Then, when an equation of “x” and “z” is established, all we need to do is rewrite all of the “z” into y/x. After simplification, the equation between “y” and “x” can be established and the problem can be solved.
It is very interesting that some concepts and techniques never go away; from day one when I took functions to today when I am taking differential equations, substitution has been a necessary technique. I started using it when I was in primary school to solve systems of equations. For example, y=3x-7 and 2x+2y=2 is a group of two-variable linear equations. The standard procedure is to substitute the second equation’s “y” as “3x”, thus creating an equation that has only one variable: 2x+2(3x-7) =2, and clearly x=2, y=-1. In calculus, the idea of u-substitution is extremely common for integration. The basic idea is to use a new variable, “u” to substitute a certain application of x to simplify the function and calculate the integral of the function in “u” and rewrite it into the application of “x” afterwards. (For more information about U-substitution click here.)
All the examples of substitution in different levels of math are eventually the same. It is a way to simplify and make something that seems extremely complex simple. These kinds of math techniques seek to establish order from chaos. After all, isn’t that eventually the essence of math?
picture of the latin phrase “much in little” from http://fmhs.mesa.k12.co.us/clubs/Multum_In_Parvo_HOME.html