Exponential Functions Student Module
1.
Write an exponential function to model each situation. Find the
value of the function after 8 years.
a.
The value of a $90,000 home increases by 4% each year.
b.
A $20,000 car depreciates at a rate of 13% per year.
FIRST ORDER LINEAR
In order to solve a linear first order differential equation, you MUST start with the differential
equation in the form shown below.
Method of Solution1. Put the differential equation in the correct initial form.
2. Find the integrating
NONHOMOGENEOUS
A second order, linear nonhomogeneous differential equation is
where g(t) is a non-zero function.
Method of Solution1. First, we will call
equation to the first equation above.
(2) the associated homogeneous differential
2. To prove that Y1
EXACT
An exact differential equation is any differential equation that applies to the following set
of rules:
Suppose that we have the following differential equation:
Note: Its important that it be in this form! There must be an = 0 on one side and the
s
HOMOGENEOUS
WITH CONSTANT COEFFICIENTS
FOR REAL, DISTINCT ROOTSMethod of Solution1. We start with the differential equation.
2. Write down the characteristic equation.
3. Solve the characteristic equation for the two roots, r1 and r2. This gives the two
HOMOGENEOUS
The first substitution well take a look at will require the differential equation to be in the form,
First order differential equations that can be written in this form are called homogeneous
differential equations.
Note: that we will usually
CAUCHY-EULER EQUATION
A Cauchy-Euler equation is where b and c are constant numbers. Let us consider the change of
variable x = et.
Then we have
The equation reduces to the new equation
We recognize a second order differential equation with constant coeff
RICCATIS EQUATION
Consider the first order differential equation
If we approximate f(x,y), while x is kept constant, we will get
If we stop at y, we will get a linear equation. Riccati looked at the approximation to the second
degree; he considered equati
BERNOULLI EQUATION
A Bernoulli differential equation is any differential equation that can be written in this
form:
Where p(x) and q(x) are continuous functions on the interval were working on and n is a
real number.
Method of Solution1. First, notice tha
SEPARABLE
A separable differential equation is any differential equation that we can write in this form:
Note: In order for a differential equation to be separable, all the y's in the differential
equation must be multiplied by the derivative and all the
Notes on Bessels Equation and the Gamma
Function
Charles Byrne (Charles [email protected])
Department of Mathematical Sciences
University of Massachusetts at Lowell
Lowell, MA 01854, USA
April 8, 2009
1
Bessels Equations
For each non-negative constant p, the
REDUCTION OF ORDER
Reduction of order can be used to find second solutions to differential equations. However,
this does require that we already have a solution and often finding that first solution is a very
difficult task and often in the process of fin