>> The last topic in mathematics that I want to cover is differential equations and transforms. And here first of all, we look at classification of differential equations then give solutions to linear homogeneous equations. Then look at Fourier series and transforms, and finally Laplace transforms. So let's start with classification of differential equations. So a differential equation has a function f(x) and some derivatives of it. And basically, we would like to solve it generally for this function. A little bit of terminology, first of all the order of the differential equation is the highest derivative. For example, dy by dx is equal to co sign x, the highest derivative there is dy by dx which is a first derivative. And therefore we say that this is a first order equation. This equation, d2y by dx squared plus 4y = 0, here the highest order of differentiation is two, second differential. Therefore we say that this is a second order equation etcetera. Ordinary differential equations contain only ordinary differentials, as opposed to partial differential equations. For example, this equation, partial d2u by dx squared + partial d2u by dy squared = 0 is a partial differential equation. But, implicitly at least, we will only be interested in the FE exam in ordinary differential equations. Linear differential equations consist of multiples of a functions and its derivatives. And in its most general form it looks like this. f(x), dny by dx squared + etcetera. Where all of the functions vex are arbitrary functions of x here. And in addition, we have an arbitrary function of x on the right hand side of the equation. So some properties of linear equations are that if we have separate solutions to the equation. Then a third solution is the sum of the individual solutions, we can simply add them together. It's a simple derivation. And a particular form of this equation, if the right hand side f(x) is 0, then we say that this is a homogeneous equation. And otherwise, if it's not a linear equation, it's a non-linear equation. But again, implicitly, we don't need to worry about non-linear equations here. I think that the exam will only discuss linear differential equations. So, here is the extract from the reference handbook. And it starts at some special cases, if the functions of x which are the multipliers of the derivatives of the left hand side, are simple constants. In other words, numbers or coefficients then the equation reduces to this, bm, dny, dx to the m plus etcetera is equal to some function of x. And furthermore if we have a homogeneous equation with constant coefficients, in other words all those values b are simple numbers, simple constants. Then we arrive at this particular equation, bn, dn, d3ny, dx to the n, etcetera is equal to 0, and this is a linear homogenous equation with constant coefficients. And this equation has a general solution as given here, y of x is = C1e, raised to the power 1x, etcetera, where all the c1, c2, are coefficients. And the rn exponents are the nth route of the characteristic equation formed in this way. P of r is equal to bn r to the n, etcetera is equal to 0. So, let me do illustrate that with by means of an example. The differential equation d2y by dx squared + 8x dy by dx + 5x is =0 is which type of equation? Which of these? Is it Linear, second order, homogeneous? Nonlinear, etcetera. Or none of the above? Well, let's start with the equation and rewrite that in its standard form. The form that we write the linear equation in, so it becomes this. d2y by dx squared + 8x dy by dx is = -5x is the equation written in this general form. Now, what kind of equation is it? Well, it's a linear equation it follows that, Its second order the highest derivative is d2y. However it is not homogeneous because the right hand side here is not equal to zero. So how about the second one? Well it is a linear equation, it's not a non-linear equation. So the answer cannot be B. How about C? Well, that one neither because it's not a partial differential equation. The only derivative's in that equation are ordinary derivative's. So, it's not that. So the answer is, rather surprisingly, none of the above. What is it? Well, it's a linear, second order, non-homogeneous equation. And this finishes our elementary discussion of differential equations, and next we will do some examples
