-----SAMPLE READING REPORT-----

5.2.  In this section Varberg et al. introduce differential equations, 
which is where we're trying to find a function whose derivatives satisfy 
some equation.  They discuss the method of separation of variables, where 
you move all the y's and dy's to one side and the x's and dx's to the 
other and integrate.  They give examples with velocity and acceleration.

Question:  They talk about derivatives and differentials.  What's a 
differential?  And I'm not at all familiar with the physics equations in 
Example 5.


7.6.  They discuss differential equations that are not separable.  A 
differential equation has a general solution, and if we specify an initial 
condition we also get a particular solution.  The trick with inseparable 
equations is to multiply by the integrating factor, which they sort of 
pull out of a hat.  They work numerous examples.

Question: Sometimes they keep the + C  when they integrate and sometimes 
they don't.  When do we need the + C and when can we ignore it?


18.1.  They throw out a bunch of confusing notation, talk about nth-order 
differential equations in general, and make reference to the sections we 
read earlier and also section 7.5, which for some reason we didn't read. 
Eventually they get to the meat of the section, which is equations of the 
form y'' + a_1 y' + a_2 y = 0.  For these we solve the auxiliary equation, 
and the answer is one of three things depending on the roots of that 
equation.

Question: What's all this business with linear operators?  That was really 
confusing.



-----ANSWERS TO HYPOTHETICAL QUESTIONS-----

> What's a differential?

When you have dy/dx, dy and dx are differentials.

> I'm not at all familiar with the physics equations in Example 5.

You won't be expected to know any physics equations, either these or 
any of the electricity stuff (e.g. Kirchhoff's Law) the examples in 7.6. 
If you need a physics equation to work a problem, it will be given to you.

> When do we need the + C and when can we ignore it?

You can drop the + C when you find the integrating factor, but when you 
integrate everything in the end you should keep the + C.

> What's all this business with linear operators?

The point is that is that the theory differential equations is a rich and 
difficult subject, and solving y'' + a_1 y' + a_2 y = 0 is just the tip of 
the iceberg.  To study the larger iceberg, it is necessary to introduce 
the language of operators, which is beyond the scope of this course. 
Briefly, though: the functions we are used to dealing with map points on 
the line (x-values) to other points on the line (y-values).  An operator 
maps functions to other functions; it is a function of functions.  For 
example, we are familiar with the derivative operator, which maps a 
function to its derivative, and also the integral operator, which maps a 
function to its integral.  Both of these are linear operators, since
(f + g)' = f' + g' and integral (f + g) = integral f + integral g.