Second Order
Equations
So Far…
We have been solving linear and
nonlinear first order equations.
Those days are
over.
(Sort
of)
Today, we will start examining
second order linear equations.
Bad (or Good?) News
Higher order equations are
MUCH more difficult to solve than
first order equations.
Linear equations
are hard.
Nonlinear equations
are crazy hard.
So we start simple,
and work our way up.
Linear Second
Order Equations
The form of a second order linear equation:
Not simple enough!
Still not simple enough!
Homogenous
equation.
Linear Second
Order Equations
The form of a second order linear equation:
Not simple enough!
Still not simple enough!
Linear Second
Order Equations
The form of a second order linear equation:
Not simple enough!
Still not simple enough!
Okay, that’s a good place to start.
Second Order Linear Homogenous
Equations with Constant Coefficients
Side note:
We already know one
specific solution to this equation.
Equilibrium Solution!
Always check for equilibrium solutions!!!!
Second Order Linear Homogenous
Equations with Constant Coefficients
We will consider solutions of the form:
Why
?
There is a
reason.
We will discuss it later in the course.
Second Order Linear Homogenous
Equations with Constant Coefficients
We will consider solutions of the form:
Second Order Linear Homogenous
Equations with Constant Coefficients
We will consider solutions of the form:
Then
:
Second Order Linear Homogenous
Equations with Constant Coefficients
We will consider solutions of the form:
Then
:
Second Order Linear Homogenous
Equations with Constant Coefficients
We will consider solutions of the form:
Then
:
Second Order Linear Homogenous
Equations with Constant Coefficients
We will consider solutions of the form:
Then
:
Second Order Linear Homogenous
Equations with Constant Coefficients
Factoring out:
One of these two must be 0
Second Order Linear Homogenous
Equations with Constant Coefficients
Factoring out:
Can’t be 0
Second Order Linear Homogenous
Equations with Constant Coefficients
Factoring out:
Must be 0
Second Order Linear Homogenous
Equations with Constant Coefficients
Factoring out:
S
o
This is called the characteristic equation.
So to solve…
Find the characteristic equation
Find
(Either factor or use the quadratic equation)
Write the general solution:
The catch…
Because there’s always a catch….
Find
Two
solutions!
or
Three possibilities
So to solve…
Find the characteristic equation
Find
and
Write the general solution:
Specific Solutions
For first order equations, need 1 initial condition
For second order equations, need 2 initial conditions
For order N equations, need N initial conditions
Specific Solution
For second order equations, need 2 initial conditions
To find specific solution:
Plug in Initial Conditions:
Specific Solution
For second order equations, need 2 initial conditions
To find specific solution:
Plug in Initial Conditions:
Solve
for:
and
So to solve…
Find the characteristic equation
Find and
Write the general solution:
Plug in Initial Conditions:
Solve
for:
and
Example
Find the characteristic equation
Find
and
and
Example
Find the characteristic equation
Find
and
and
Write the general solution:
Plug in Initial Conditions:
Solve
for
and
and
Example
Find the characteristic equation
Find
and
and
Write the general solution:
Plug in Initial Conditions:
Solve
and
for
and
Summary
•
Higher Order Equations are Difficult To Solve
•
Can solve Second Order Homogeneous Linear
Equations with Constant Coefficients Using
Characteristic Equations
•
Need 2 Initial Conditions for Second Order
Equations
•
In general, need N initial conditions for Nth order
equation
Questions?