Gavin's DiffEq Class Clarification: Jan 24
Question: What was that about nullclines?
The book reports, in slightly different words, that a nullcline is
a line where solutions to the differential equation have zero
. The book actually says that nullclines are ``curves of
zero inclination''---notice that this is the same thing. So, if we
want to find the nullclines for a differential equation
y'(x) = f(x,y),
we have to find where f
) = 0. Let's
look at an example: Suppose
y'(x) = x2 - y.
Then nullclines are where x2
= 0, so y
(This gives only one nullcline.) Notice that this isn't a solution,
because it would have to have to satisfy the differential equation:
) = 2x
, so if we plug into the
differential equation we get
2x = x2 -
x2 = 0,
which isn't true for all x
, so y
isn't a solution to the differential
However, if we had found that a nullcline
that was a constant, it would also be a solution curve for an
Question: How do I do substitution to integrate?
Let's do an example: Let's suppose that Intab
is the integral from a
, for the purposes of
notation. The integral we'll do is
Int[ (2x - 1) sin(x2 - x) ]dx.
We notice that there is a function composition here (function in
another function): the x2
appears inside the sine function. So let's let
. This turns the integral into
Int[ (2x - 1) sin(w) ]dx.
Then we need to get rid of the dx
in the integral.
, we get
- 1, so
- 1) dx
. And as luck
would have it, that's exactly what's left in the integral! So it
Int[ sin(w) ]dw = -cos(w) +
C = -cos(x2 -
x) + C.
Of course, this is a really simple example: possible complications
- if the integrand has a multiple of the differential dw:
then we will get a constant times the answer, or
- if the integrand has left-over x's after we've
substituted for the function w and dx: in this
case, we might have to solve the equation for w to get
x = g(w) and substitute the
g(w) to get rid of the x's---Note that
we can't have any x's left over in the equation after
finishing the substitution, or
- if the function to choose for w isn't so obvious. In
this case, try (1) any function inside another function, or
(2) any function inside a square root, raised to a power,
or in the denominator of a fraction, or (3) some other
A Calc II clarification
gives another take on substitution.
Gavin's DiffEq Clarification 000124
Last Modified: Mon Jan 24 23:32:22 CST 2000