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Differential Equation
The first definition that we
should cover should be that of differential equation. A
differential equation is any equation which contains derivatives, either
ordinary derivatives or partial derivatives.
There is one differential equation
that everybody probably knows, that is Newton’s
Second Law of Motion. If an object of mass m is moving with acceleration a
and being acted on with force F
then Newton’s
Second Law tells us.
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(1)
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To see that this is in fact a
differential equation we need to rewrite it a little. First, remember that we
can rewrite the acceleration, a,
in one of two ways.
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(2)
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Where v
is the velocity of the object and u
is the position function of the object at any time t. We should also remember at this point that the force, F may also be a function of time,
velocity, and/or position.
So, with all these things in mind Newton’s Second Law can
now be written as a differential equation in terms of either the velocity, v, or the position, u, of the object as follows.
So, here is our first differential equation. We will see both forms of this in later
chapters.
Here are a few more examples of differential equations.
Order
The order of a
differential equation is the largest derivative present in the differential
equation. In the differential equations listed above (3) is
a first order differential equation, (4), (5),
(6),
(8),
and (9)
are second order differential equations, (10) is
a third order differential equation and (7) is
a fourth order differential equation.
Note that the order does not
depend on whether or not you’ve got ordinary or partial derivatives in the
differential equation.
We will be looking almost
exclusively at first and second order differential equations in these
notes. As you will see most of the
solution techniques for second order differential equations can be easily (and
naturally) extended to higher order differential equations and we’ll discuss
that idea later on.
Ordinary and Partial Differential Equations
A differential equation is called
an ordinary differential equation, abbreviated by ode,
if it has ordinary derivatives in it. Likewise, a differential equation is
called a partial differential equation, abbreviated by pde,
if it has differential derivatives in it. In the differential equations above (3)
- (7)
are ode’s and (8)
- (10)
are pde’s.
The vast majority of these notes
will deal with ode’s. The only exception
to this will be the last chapter in which we’ll take a brief look at a common
and basic solution technique for solving pde’s.
Linear Differential
Equations
A linear differential
equation is any differential equation that can be written in the following
form.
The important thing to note about linear differential
equations is that there are no products of the function, ,
and its derivatives and neither the function or its derivatives occur to any
power other than the first power.
The coefficients and can be zero or non-zero functions, constant or
non-constant functions, linear or non-linear functions. Only the function, ,
and its derivatives are used in determining if a differential equation is
linear.
If a differential equation cannot be written in the form, (11)
then it is called a non-linear
differential equation.
In (5) - (7)
above only (6)
is non-linear, all the other are linear differential equations. We can’t classify (3) and
(4)
since we do not know what form the function F
has. These could be either linear or
non-linear depending on F.
Solution
A solution to a
differential equation on an interval is any function which satisfies the
differential equation in question on the interval . It is important to
note that solutions are often accompanied by intervals and these intervals can
impart some important information about the solution. Consider the following example.
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Example 1 Show
that is a solution to for .
Solution We’ll need the first and second
derivative to do this.
Plug these as well as the function into the differential equation.
So, does satisfy the differential equation and
hence is a solution. Why then did I
include the condition that ? I did not use this condition anywhere in
the work showing that the function would satisfy the differential equation.
To see why recall that
In this form it is clear that we’ll need to avoid at the least as this would give division by
zero.
Also, there is a general rule of thumb that we’re going to
run with in this class. This rule of
thumb is : Start with real numbers, end with real numbers. In other words, if our differential
equation only contains real numbers then we don’t want solutions that give
complex numbers. So, in order to avoid
complex numbers we will also need to avoid negative values of x.
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So, we saw in the last example
that even though a function may symbolically satisfy a differential equation,
because of certain restrictions brought about by the solution we cannot use all
values of the independent variable and hence, must make a restriction on the
independent variable. This will be the
case with many solutions to differential equations.
In the last example, note that there are in fact many more
possible solutions to the differential equation given. For instance all of the following are also
solutions
I’ll leave the details to you to check that these are in
fact solutions. Given these examples
can you come up with any other solutions to the differential equation? There are in fact an infinite number of
solutions to this differential equation.
So, given that there are an infinite number of solutions to
the differential equation in the last example (provided you believe me when I
say that anyway….) we can ask a natural question. Which is the solution that we want or does it
matter which solution we use? This
question leads us to the next definition in this section.
Initial Condition(s)
Initial Condition(s)
are a condition, or set of conditions, on the solution that will allow us to
determine which solution that we are after.
Initial conditions (often abbreviated i.c.’s when I’m feeling lazy…) are
of the form,
So, in other words, initial conditions are values of the
solution and/or its derivative(s) at specific points. As we will see eventually, solutions to “nice
enough” differential equations are unique and hence only one solution will meet
the given conditions.
The number of initial conditions that are required for a
given differential equation will depend upon the order of the differential
equation as we will see.
Initial Value Problem
An Initial Value
Problem (or IVP) is a
differential equation along with an appropriate number of initial conditions.
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Example 3 The
following is an IVP.
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Example 4 Here’s
another IVP.
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As I noted earlier the number of initial condition required
will depend on the order of the differential equation.
Interval of Validity
The interval of
validity for an IVP with initial condition(s)
is the largest possible interval on which the solution is
valid and contains . These are easy to define, but can be
difficult to find, so I’m going to put off saying anything more about these
until we get into actually solving differential equations and need the interval
of validity.
General Solution
The general solution
to a differential equation is the most general form that the solution can take
and doesn’t take any initial conditions into account.
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Example 5 is the general solution to
I’ll leave it to you to check that this function is in fact
a solution to the given differential equation. In fact, all solutions to this differential
equation will be in this form. This is
one of the first differential equations that you will learn how to solve and
you will be able to verify this shortly for yourself.
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Actual Solution
The actual solution
to a differential equation is the specific solution that not only satisfies the
differential equation, but also satisfies the given initial condition(s).
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Example 6 What
is the actual solution to the following IVP?
Solution This is actually easier
to do that it might at first appear.
From the previous example we already know (well that is provided you
believe my solution to this example…) that all solutions to the differential
equation are of the form.
All that we need to do is determine the value of c that will give us the solution that
we’re after. To find this all we need
do is use our initial condition as follows.
So, the actual solution to the IVP is.
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