# Engineering Mathematics I

1. **The Second Mid-Term** covers Chapter 9 (nonlinear
differential equations

and stability), Chapter 6 (Laplace Transform), and Chapter 5

(Series Solutions). Materials previously covered in Mid-Term #1 are

assumed known (e.g. direction fields, undetermined coefficients, integration

factor, . . . ).

2. **Chapter 9, Nonlinear differential equations and stability.** This

is the fun chapter. The basic ideas are presumably good for more than

two variables , but the most successful applications of the basic ideas

are for two variable problems . (Remember, there is no chaos in two-dimensional

problems).

• What does “autonomous” mean? Do you know how to make an

nonautonomous problem into an autonomous problem? (ask me

in class).

• What is meant by “stability” of an equilibrium point? What is

asymptotic stability?

• Given a two-dimensional problem dx/dt = F(x, y), and dy/dt =

G(x, y). You should be be pretty comfortable at sketching direction

fields (called “phase plane” sketch here). Use the trick I

taught you (make sure you know whether the sign of F (or G)

changes (or does not change ) across the F = 0 (or G = 0) line(s)).

• Once you have located the “crossing points” of the F = 0 line(s)

with the G = 0 line(s), you have located the singular points.

(Note: sometimes F = 0 has more than one lines, and the crossing

points among themselves are not singular points. The same

remark applies to G = 0). The phase plane graphical sketch can

give you a very good hint on what kind of singular point(s) you

are dealing with. To pin your conclusion down, you need to do

more work.

• How do you study the properties of a singular point? You look

at it under a microscope, of course! Mathematically , this means

you expand F(x, y) and G(x, y) about the singular point (say
)

using a Taylor Series expansion, keeping only the lowest order

terms (because and
are very small).

• The Taylor series of F(x,y) about
is:

If is a singular point, the first term is zero . The
terms

represented by . . . are “higher order” and are negligible (if

is small, then is smaller!). In other words: any smooth

curve under a microscope looks like a straight line!

• Note: Often, you don’t need to compute all those derivatives to

get your Taylor Series. For example, synthetic division is often

an easier way to do things (what is the Taylor series of 1/(1 − x)

about x = 0.2?).

• So, any 2-dimensional singular point under a microscope can be

looked at in the phase plane:

where a, b, c, e are constants and the . . . are terms we
are neglecting

under the microscope. You are expected to be able to fully discuss

a singular point with given values of a, b, c, e, using information

about the eigenvalues and eigenvectors.

• With the . . . terms neglected, you can make statements
about the

stability of the original system using the local linearized analysis.

Your linear conclusions are applicable to the case when the . . .

terms are not summarily neglected EXCEPT for the case of purely

imaginary eigenvalues. You should know why this special case

needs special attention.

• Limit cycles! What is a limit cycle? What are tell tale signs that

there is a limit cycle lurking in your two-dimensional phase plane

sketch?

3.** Chapter 6, Laplace Transform.** You will be given a xeroxed copy

of the table on page 300.

•You must know the definition of Laplace transform. Even though

the table on page 300 is given to you, I will ask you to work out

some of them long hand.

•You need to refresh your partial fractions. You will need it.

•You need the convolution integral which is item #16 on the table.

•You need to know how to use the step function
to extract

parts of a given function f (t).

•You need to know the Dirac Delta function
and how to

compute its Laplace transform.

•You must know how to evaluate the following integral

for t < 17.85 and for t > 17.85.

4. **Chapter 5. Series Solutions of Second Order Equations .** The

inspired guess is: the solution near the point
is a power series of

! The attention here is limited to linear, second order ODE’s:

Without loss of generality, we set .

• If P(0) ≠ 0, then x = 0 is an ordinary point. The power
series

is a Taylor series. The only trick you need to practice is how to

shift the index of summation.

• If P(0) = 0, then x = 0 is a singular point. There are two kinds

of singular points: regular and irregular.

• You need to know how to identify a regular singular point (use

the definition given in the book). Once it is identified as a regular

singular point, an inspired guess for the solution is the product of

x ^{r} and an innocent ordinary-looking Taylor series. Here r is called

the exponent of the regular singular point. It is determined by the

indicial equation which is a quadratic (for second order ODE’s).

• The mechanics of working such problems is straight forward: use

the trick of shifting the indices of summation to get “all” the

terms under a single summation sign with the same x^{n} factor, any

“left over” terms are then written out explicitly. Setting each of

the leftover terms to zero gives you the “indicial equation,” and

explicit values of some early coefficient. Setting the coefficient of

x ^{n} in your big summation term gives you the “recurrence relation.”

Viola, its done.

• If and
is not an integer, then you are in luck.

You need to be able to do such problems, getting two linearly

independent solutions using and
one after the other.

• If or
is an integer (both and
are real), you

need to know there is a problem, and you need to know where to

go in Boyce and Diprima to find the inspired guess for the second

solution.

• We did not cover irregular singular point. If you are confronted

with such a singular point, you are more or less on your own to get

your inspired guess. A good inspired guess is: try an exponential

factor in place of x^{r} and see if it works.

Good luck!

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