Download differential equation

Modeling
with Differential
CHAPTER
2
2.4Equations
Continuity
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 ).
By Newton’s CHAPTER
Second Law, we have:
2
2x /d t2 = - k x.
m
d
2.4 Continuity
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.
A function f is
called
a
solution
of
a
differential
CHAPTER
2
equation if the equation is satisfied when
y = f(x) and2.4
its derivatives
are substituted in the
Continuity
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).
Example: Verify that y = (2 + ln x) / x is a solution
CHAPTER
2
2
of the initial-value problem x y’ + x y = 1
y (1) = 2. 2.4 Continuity
Example: For what values of r does the function
y = e r t satisfy the differential equation
y’’ + y’ – 6y = 0 ?
Example: Show
that
every
member
of
the
family
of
CHAPTER
2
2 /2
x
functions y = Ce
is a solution of the
2.4 Continuity
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.
Example: A function y(t) satisfies the differential
CHAPTER
equation dy/dt
= y4 – 6y3+ 5y22
.
2.4
Continuity
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?
Direction Fields
CHAPTER
2
2.4a Continuity
Suppose we have
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.
Example: Sketch a direction field for the
differentialCHAPTER
equation y’ = x y +2
y2. Then use it to
sketch three2.4
solution
curves.
Continuity
Example: Sketch the direction field for the
2Then use it to
differentialCHAPTER
equation y’ = x2 + y.
sketch a solution
curve that passes through (1,1).
2.4 Continuity