Introductory Concepts Chapter 1 Introductory Concepts

Sec 1.2 – Basic Concepts A differential equation is an equation that relates an unknown function and one or more of its derivatives So if y = f(x), we can write for example Usually y is the function with x its independent variable (motion in the xy-plane), or x is the function with t its independent variable (particle motion along the x-axis with respect to time)

Examples .

Solutions of DE’s A solution of a DE is a function y(x) (or x(t)) that is continuous on an interval on the x-axis (t-axis) Satisfies the equation in question The interval can be found, usually by looking at the equation and seeing where everything’s defined.

Solutions of DE’s Some solutions can be found just by inspection Example: Find a solution y(x) to the equation

Solutions of DE’s Some are tougher, but can be verified Example: Verify that is a solution to the differential equation

Notation and Terminology The order of a DE is the highest derivative of the solution function that appears. The general form of an nth-order differential with unknown function y of independent variable x is So there can be terms involving x (the independent variable), y (the function itself), y’ (its first derivative), and so on You can now try problems 1 and 2 in Sec 1.2

You’ve already seen some DE in Calc “The rate of change of a population is proportional to its size” Find functions for the velocity and altitude of a projectile fired upward from ground level with initial velocity 100 ft/sec.

Integrals; Particular Solutions Suppose that the derivative of the function y(x) that we seek depends only on x’s, and not on y. IOW, Then

Examples 1. Then = For what interval on the x-axis is this solution valid? So there are an infinite number of solutions. How can we find a particular one?

Example, with initial value Suppose we know that And that when x=1, y=3. We can now find C, and write a particular solution.

Another example of an initial value problem Find a solution to the IVP

2nd Order Equation General: Then y’= So y= And we need two initial conditions to find

2nd Order Example Ex: Then

Common 2nd-order DE– a(t), v(t), x(t) As we know, Thus As we have seen, in order to find a particular x(t), we will need an initial velocity and an initial position In the particular case where the acceleration is constant a, After that, just need to line up the units and ID the question(s) being asked.

Example A ball is thrown straight down from a building w initial speed of 10 m/sec. It hits with speed 60 m/sec. How tall is the building?

Example Find y, if Now do the rest of the Sec 1.2 problems on the schedule

Example Find y, if y= Let Then