Class Notes (835,598)
Mathematics (269)
MATH 310 (27)
Lecture

# Note-1.pdf

3 Pages
140 Views

School
Department
Mathematics
Course
MATH 310
Professor
Roberto Armenta Barrera
Semester
Fall

Description
Note #1 MATH 310 Introduction to Ordinary Di▯erential Equations CLASSIFICATION OF ORDINARY DIFFERENTIAL EQUATIONS What is a di▯erential equation? It is an equation that establishes a functional relationship between one independent variable and one dependent variable. For example, let us assume that the dependent variable is y and that the independent variable is t. The equation dy + ln(t)y = 1 (1) dt establishes a functional relationship y = y(t) for t > 0. How do we ▯nd y(t)? That is the central question of this course. To discuss di▯erential equations, it is ▯rst necessary to de▯ne some vocabulary that allows us to classify di▯erential equations. Order of a Di▯erential Equation As before, let us assume that the dependent variable is y and that the independent variable is n n t. A derivative of order n is usually written in the following manner: d y=dt . The order of a di▯erential equation is the same as the order of the highest derivative that appears in the equation. Equation (1) is thus a ▯rst-order equation. Ordinary vs. Partial Di▯erential Equation If the di▯erential equation (or equations) contain only one independent variable, then we say that the di▯erential equation (or equations) are ordinary. If the di▯erential equation (or equations) con- tain more than one independent variable, then we say that the di▯erential equation (or equations) are partial. One example of a partial di▯erential equation is the transport equation: @u @u @t + [email protected] = 0 where u = u(x;t) and a = constant: (2) An example of a ▯rst-order ordinary di▯erential equation was given earlier in (1). Systems of Ordinary Di▯erential Equations If were to have a collection of m dependent variables, we would need m di▯erential equations to completely specify the m dependent variables. If m > 1, we say that we are dealing with a system of di▯erential equations. For example, the system of two coupled equations dv + aw = sin(t) where a = constant (3) dt and dw dt + bv = t + 1 where b = constant: (4) speci▯es the two functions w(t) and v(t). Because there is only one independent variable, this is a system of ordinary rather than partial di▯erential equations. Fall 2013 Page 1/3 MATH 310 Introduction to Ordinary Di▯erential Equations Note #1 Linearity A function f(t) is said to be linear if it satis▯es the following property: f(▯t) = ▯f(t) for any constant ▯: (5) In other words, if we were to make the substitution t ! ▯t in f(t), the result should be the same as what we would get if we muiltiplied f(t) by ▯. The concept of linearity applies in a slightly di▯erent way to equations. The test for linearity is de▯ned in terms of the dependent variable rather than in terms of the independent variable. Suppose that y is the dependent variable and t is the independent variable. To establish the linearity of a di▯erential equation that speci▯es y = y(t); we must ▯rst rewrite the di▯erential equation the following form: ▯ n ▯ F t;y;dy;:::;d y = G(t) (6) dt dtn where F contains all the terms that are dependent on y and G(t) contains all the te
More Less

Related notes for MATH 310
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.