# Ordinary Differential Equations and Linear Algebra

**1. Separable Equations Revisited **

Good Morning, everyone. I hope you all are feeling good about ODE. It is

one of the most fascinating subject of math I have ever studied. You have seen

many applications of ODE, we will see more along the way. Please let me know

if the pace is good for you.

Last time, we started talking about separable equations. It is the easiest

form of ODE you will encounter in this course. We will finish the rest of the

examples from last time.

Also you might be taught before that in , dy
and dt are both just symbols ,

they are not really meaningful . But in that case, how can you separate them

also?

Here is another interpretation for way of solving separable equation.

Given the separable equation, , we have

integrating both side, we get

Notice that LHS is just a change of variable from ttoy . So
the ODE is then

given by Today we will be talking about
another several

types of ODEs. We will start from linear equations .

**2. First Order equation**

**Definition: **A first-order equation is one of the
form

if f(t) ≡ 0, the equation is called homogeneous. Otherwise it is
inhomogeneous.

Sometimes, the coefficient of x ' is not a constant, or not 1, i.e. it is of a
more

general form

However, we should notice here the coefficient of x' and x are both in terms

of t only, the term like x ^{2}, xx' and cos x is not allowed by the
definition of first

order equation. Here are several examples for you to judge whether they are

first order or not.

Now we want to know if there is some general way to solve
the first order

equation. We start from the homogeneous ones.

We can do this in the very general setting as follows:

Given d, move the x terms to the left , and t
term to the right as

we did in the separable equations. (this IS separable equation itself.)

we have

so

exponentiating, we get

You can either keep this formula in mind or redo the above
steps when you see it.

Here are a few practice examples.

(this is not a
homogeneous equation)

From the above last problem, we see some inhomogeneous equation. This is

easy because it is also a separable equation. As the f(t) is a constant. We will

see the inhomogeneous equation could also be solved in general settings.

This is what is called the integrating factor method.

Namely, we introduce a integrating factor, we call it the

We then multiply both sides of the equation as follows.

So then we solved the

I have taught this method before . The feeling my students
had is this is

not very natural. Yes, I admit this is a very creative and constructive idea.

Everything comes into such a beautiful form. What we can do here is just to

appreciate the beauty of math.

There are a few points you have to pay attention here.

1. We solve this in the general form

That means if you want to memorize the above formula, you will need to write

the given equation into the above form, and find what your a(t) and f(t) is.

2. As we did in studying calc II, we should not forget about the constant C.

Here the constant C is considered before you really write out the answer. That

would be a big mistake if you miss the constant.

3. However when we compute the integrating factors, we kind of need a simple

version of the many possible factors. So we just need to find ONE possible

integrating factor, without involving an extra constant in the integrating
factor.

If you have the time and interests, you can try to verify
the integrating factor

with different choices of C’s could not
affect your answer at all.

Here are a few examples to make you familiar with integrating factor method.

1. y' = −2y + 3 Here use 2 ways to find the solutions and check if they are

the same.

2.

Solve:

a(t) = 1, and f(t) = e^{-t}. The integrating factor.

So according to our formula

3. You can of course combine the initial value problem into the first order

equations.

Find the solution of

that satisfies x(0) = 1

4. find the general solution of x' = x tan t + sin t and also find the solution

that satisfies x(0) = 2.

**Solve:**

a(t) = tan t, and f(t) = sin t. The integrating factor is given by u(t) =

So we have So

5. In class practice problem:

**3. Structure of the solutions of first order ODE**

Now you all have some feelings about this beautiful idea
about integrating

factors. You may wonder why you distinguish the homogeneous and inhomogeneous

case. Are there any connections between these 2 cases. They are both

linear equations though.

You will see that more clear when you see matrix algebra . But now you can

still get a glimpse of the idea.

Look at the homogeneous equation

the solution is given by

where the solutions for the inhomogeneous ones

are given by

The underlined part is the solution of the homogenenous
siblings.

Moreover, any solution of the inhomogeneous equation is the sum of a solution

from the homogeneous equation and a fixed special solution from the

inhomogeneous. i.e the solution of the inhomogeneous equation is given by

where is the solution of the homogeneous
equation, where the is a

particular solution of the inhomogeneous equation.

**3. Variation of Parameters:
**

Idea: you can always write

**Example:**y' = −2y + 3

**4. Model of Motion:**

Model of Motion with air resistence. The basic model is that the force of air

resistence is given by R(x, v) = −r(x, v)v where r is a nonnegative function.

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