MATH 128 Introduction to Differential Equations (FCV)

Introduction to Differential Equations:



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1. Ok, so we are getting started with differential equations. What are differential equations? They look something like this: y'+y=8-x or figure 1. By simply looking at that example, the very first difference anyone should notice between this new differential equation and a regular equation is that one of the variables have a "prime" symbol. So, we are no longer looking at just a function with unknown variables, we are looking at a function with unknown variable(s) and its derivative!
* Since variables can be expressed in terms of a function (ie. y=f(x) and x=f(y)), we can also define differential equations as an equation that involves unknown function(s) and its derivatives.

2. To solve differential equations, or find this unknown, integration will be used to "undo" the derivative. A constant of integration "+C" will be incorporated if the exact solution needs to be found. 

Differential equations are any equation that involves an unknown function and its derivatives.

Figure 1

3. A second order differential equation is simply an equation with a 2nd order differentiated variable. So, f''(x)+f'(x)=x would be a 2nd order, and x+y+y'''=23 would be 3rd order... you get the pattern

4. Differential equations can also be classified into linear and non linear. Linear differential equations are when the derivatives (of any order) are to the power of 1, while non linear differential equations are when any one of the derivatives (of any order) are to a power of more than 1.
Linear: y'+y'''''=0    or     y^5+y'+4y''+3x^3=0   or    y''-y'+y^2=4x
Nonlinear: y'^2+y=x    or    2y''+ 5y'^2+y=x^2   or    y'''+y'^2+y=x

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