Transcript 11.1 Differential Equations Intro

Section 11.1
What is a differential equation?
• Often we have situations in which the rate of
change is related to the variable of a function
• An equation which gives information about the
rate of change of an unknown function is
called a differential equation
• Differential equations have functions as
solutions
– As opposed to numbers (solutions of algebraic
equations)
• Differential equations model more complex
problems where the solution is described by a
function
Example
• A yam is placed inside a 200ºC oven. The yam
gets hotter at a rate proportional to the
difference between its temperature and the
oven’s temperature. When the yam is at
120ºC, it is getting hotter at a rate of 2º per
minute. Write a differential equation that
models the temperature, T, of the yam as a
function of time, t.
dT
 0.025(200  T )
dt
T  200  Ce
0.025t
• C is an arbitrary constant
• In order to solve for C we must be given some
kind of initial condition
• The C in our case is the initial temperature
difference between the yam and the oven
• What is C if the initial temperature of the yam
is 20º?
T  200  180e
0.025t
Family of solutions for different values of C
• Antidifferentiation is actually a particular case
of solving a differential equation, in particular
dy
 f (x )
dx
• Where the constant appears added to the
solution, not multiplied
• The solution is a function and unless specific
conditions are given there are usually many
solutions to a certain differential equation
• The way to tell if a given function is a solution
to a certain differential equation is by
substitution
Examples
• Verify that the following are solutions to the
given differential equation
1)
y  2 x  4 is solution to
 x  12 y
dy
dx
2)
y  sin( 2t ) is solution to
d2y
dt
2
 4y  0
Second-order differential equations
• A second-order differential involves the
second derivative
• This involves two antidifferentiations so it will
involve two arbitrary constants
• For example let’s look at the second-order
differential equation
d2y
dx
2
x