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Welcome to calculus. I'm Professor Greist.

Â We're about to begin lecture 23. Bonus material.

Â In our main lesson, we gave a collection of substitutions involving trigonometric

Â or hyperbolic trigonometric functions. These work great for computing some

Â integrals. But at times, you've got to be careful.

Â Let's consider a simple example of a differential equation mode that is

Â amenable to a trigonometric substitution. Let's say that a financial model projects

Â that your marginal profits Are going to be a constant plus some term that is

Â proportional to the square of the net profits.

Â What do you think about such a model? Does that sound like a good investment?

Â Well let's translate what that model says.

Â Into a differential equation. If P as a function of t is your net

Â profit. Then what do we mean when we say that the

Â marginal profits are equal to a constant plus a term proportional to the square of

Â the net. Well the marginal profit.

Â Is the derivative, dP dt, to say that that marginal profit is equal to a

Â constant, let's make sure that it's a positive constant by calling b squared.

Â Plus some term proportional to the square of the net profits, let's call that term

Â a squared p squared, so that the constant of proportionality here is also positive

Â as it must be. Then how do we solve this differential

Â equation. If we perform substitution.

Â Then, on the left, we need to compute the integral of dP over b squared plus a

Â squared P squared. On the right, we have the integral of dt.

Â Now, which trigonometric substitution is appropriate?

Â Well, clearly, using a tangent is what we're going to want.

Â So, let us substitute for P, B over A times tangent of theta.

Â DP is therefore a secant squared. Now, when we plug that in, well, we get B

Â over A secant squared theta for DP. And in the denominator b squared secant

Â squared Theta. The secant squareds cancel, the b's

Â cancel, and we're left with d Theta over a times b.

Â Now that integrates to simply Theta over ab.

Â Setting that equal to the integral of a dt, that is, t plus constant, allows us

Â to very easily substitute back again for theta the arctangent.

Â We therefore have t plus a constant equals one over ab Times the arctan of a

Â over, times P. Solving for P as a function of t, after a

Â little algebra, yields b over a times tangent of quantity, a b t plus a

Â constant. We can show based on an initial condition

Â that that constant is say, zero. Let's say we're starting off with no

Â profit at all, times zero, very good. We've got our answer, and we've done the

Â mathematics correctly... But what do you think about this model

Â for the net profit? Is it realistic?

Â 4:05

Well, what is happening to P as T increases from zero?

Â Well, we're getting a tangent function which means that at some finite time, t

Â and the function is going to get a large, a very large, it's going to blow up to

Â infinity. In fact, these are called blow-up

Â solutions, or singularities, and they arise, sometimes, when performing a

Â trigonometric substitution. Based on the nature of the tangent,

Â cotangent, secant, and cosecant functions.

Â Now you may think that such a solution to a differential equation is inherently

Â nonphysical. We don't have things going to infinity

Â and finite time, however there are examples of physical settings where blow

Â ups, or singularities in the differential equation are important to consider.

Â For example in certain areas of physics one has a resonance phenomenon.

Â Think of a vibration that is forced at a certain critical frequency that leads to

Â feedback. This corresponds to a blow-up solution to

Â the relevant differential equation. But it is not something ignorable.

Â Indeed one often has to engineer around such resonances, for safety purposes.

Â Other examples include vortices, in fluid dynamics, or electromagnetics.

Â These can arise as singular solutions. To the relevant differential equations.

Â And nobody would argue that things like tornadoes or hurricanes are unimportant

Â or uninteresting. In fact, the differential equation that

Â models three dimensional fluid dynamics, that is the Navier Stokes equation.

Â Is a very fascinating set of equations, unfortunately we don't have enough

Â calculus down as yet for us to be able to appreciate these equations fully in this

Â class. However, it is an open problem in

Â mathematics, whether or not the solutions do these differential equations have

Â finite time blowups or singularities in them?

Â This is one example of a very simple sounding problem which is nevertheless

Â open and unknown to the best of mathematicians.

Â Many very good mathematicians are currently, working on this problem.

Â After you've had a little bit of multi variable calculus, you might be able to

Â work it too.

Â