CHAPTER 2 2.4 Continuity

Modeling with Differential Equations

Models of Population Growth: One model for the growth of population is based on the assumption that the population grows at a rate proportional to the size of the population.

The variables in this model are: t = time (the independent variable) P = the number of individuals in the population (the dependent variable).

The rate of growth of the population is proportional to the population size: ( dP / dt ) = k P

and any exponential function of the form P(t) = Ce k t.

( dP / dt ) k P if P is small ( dP / dt ) < 0 if P > K.

Then we can conclude that ( dP / dt ) = k P ( 1 – ( P / K ).

A Model for the motion of a Spring:We consider the motion of an object with mass m at the end of a vertical spring. Hooke’s Law says: restoring force = - k x ( where k is a spring

constant ).

CHAPTER 2 2.4 Continuity

By Newton’s Second Law, we have:

m d 2x /d t2 = - k x.

This is an example of what is called a second – order differential equation because it involves second derivatives.

General Differential Equations: In general, a differential equation is an equation that contains an unknown function and one more of its derivatives. The order of a differential equation is the order of the highest derivative that occurs in the equation.

CHAPTER 2 2.4 Continuity

A function f is called a solution of a differential equation if the equation is satisfied when y = f(x) and its derivatives are substituted in the equation.

When applying differential equations, we are usually not as interested in finding a family of solutions (the general solution) as we are in finding a solution that satisfies some additional requirement such as a condition of the form y(to) = yo (an initial condition).

CHAPTER 2 2.4 Continuity

Example: Verify that y = (2 + ln x) / x is a solution of the initial-value problem x2 y’ + x y = 1 y (1) = 2.

Example: For what values of r does the function y = e r t satisfy the differential equation y’’ + y’ – 6y = 0 ?

CHAPTER 2 2.4 Continuity

Example: Show that every member of the family of functions y = Ce x2 /2 is a solution of the differential equation y’ = x y.

Example: Verify that all the members of the family y = ( c - x2)-1/2 are solutions of the differential equation y’ = x y3.

CHAPTER 2 2.4 Continuity

Example: A function y(t) satisfies the differential equation dy/dt = y4 – 6y3+ 5y2.

a) What are the constant solutions of the equation?b) For what values of y is y increasing?c) For what values of y is y decreasing?

CHAPTER 2 2.4 Continuity

Direction Fields

Suppose we have a first-order differential equation of the form y’ = F (x,y). The differential equation says that the slope of a solution curve at a point (x,y) on the curve is F (x,y).

Short line segments with slope F (x,y) at several points (x,y) form a direction field or slope field.

CHAPTER 2 2.4 Continuity

Example: Sketch a direction field for the differential equation y’ = x y + y2. Then use it to sketch three solution curves.

CHAPTER 2 2.4 Continuity

Example: Sketch the direction field for the differential equation y’ = x2 + y. Then use it to sketch a solution curve that passes through (1,1).