Gavin's DiffEq Class Clarification: Jan 24

The book reports, in slightly different words, that a nullcline is a line where solutions to the differential equation have zero slope. 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(x,y) = 0. Let's look at an example: Suppose y'(x) = x² - y. Then nullclines are where x² - y = 0, so y = x². (This gives only one nullcline.) Notice that this isn't a solution, because it would have to have to satisfy the differential equation: here y'(x) = 2x, so if we plug into the differential equation we get 2x = x² - x² = 0, which isn't true for all x, so y = x² isn't a solution to the differential equation.

However, if we had found that a nullcline that was a constant, it would also be a solution curve for an equilibrium solution.

Let's do an example: Let's suppose that Int_a^b[ ] is the integral from a to b, 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 - x appears inside the sine function. So let's let w = x² - x. This turns the integral into Int[ (2x - 1) sin(w) ]dx. Then we need to get rid of the dx in the integral. Differentiating w, we get dw / dx = 2x - 1, so dw = (2x - 1) dx. And as luck would have it, that's exactly what's left in the integral! So it becomes Int[ sin(w) ]dw = -cos(w) + C = -cos(x² - x) + C.

Of course, this is a really simple example: possible complications could include

• 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 logical function.

A Calc II clarification gives another take on substitution.