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## UCT MAM1000 lecture notes part 37 – differential equations part vi – second order differential equations

Second Order differential equations

We are only going to look at a particular subset of all possible second order differential equations (that is, equations which contain at most second derivatives) but these particular equations are absolutely ubiquitous across every field of science. The particular subset we are going to look at are linear, homogenous second order differential equations with constant coefficients. These can be written in general as:

$\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c y=0$

It is linear because it contains at most (and in this case at least) a single power of $y$ in each term. It is homogenous because there is no term which has no powers of $y$ (ie. the right hand side is not a constant), and the coefficients $b$ and $c$ are any real numbers (though you can extend this to having complex numbers very easily). We will see that depending on the relationship between these numbers ($b$ and $c$) we can have very different behaviour of the equation.…

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## UCT MAM1000 lecture notes part 36 – differential equations part v – first order differential equations

First order linear differential equations

We are now going to deal with another subset of first order differential equations which in some ways are easier than the previous and in other ways more complicated. These are linear first order differential equations. The general form of a first order linear differential equation is:

$\frac{dy}{dx}+P(x)y=Q(x)$

where $P(x)$ and $Q(x)$ are any functions of $x$.

Very importantly, I’m leaving off the fact that $y$ is dependent on $x$ in the notation, but you should remember that this is really $y(x)$ and that is the function you are trying to solve for.

Sometimes you will be given an equation which is not obviously in this form but it can be transformed to this form. For instance:

$\frac{1}{y}\frac{dy}{dx}=x^2+\frac{\sin x}{y}$

This can easily be transformed into the canonical form for a linear first order DE. We are going to try and rewrite the left hand side of the equation in a form which will mean that we can solve the differential equation very easily.…

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## UCT MAM1000 lecture notes part 35- differential equations part iv – separable differential equations

Separable differential equations
In some ways these are the easiest differential equations to solve in theory, though in practice the final step (that of integrating) may be difficult or impossible. A separable differential equation is one of the form:

$\frac{dy}{dx}=\frac{f(x)}{g(y)}$

where $f(x)$ and $g(y)$ are any functions of $x$ and $y$ respectively. For instance:

$\frac{dy}{dx}=x y$

is of this form where $f(x)=x$ and $f(y)=\frac{1}{y}$. The reason that these equations are simple in theory is because we can rearrange them to be:

$g(y)dy=f(x)dx$

ie. we have all the $x$ stuff on one side and all the $y$ stuff on the other and then we can integrate both sides:

$\int g(y)dy=\int f(x)dx$

and that’s it. As long as you can do the integrals, you can get a function $y$ in terms of $x$. let’s look at some examples:

$\frac{dy}{dx}=x y$

gives the following integral:

$\int\frac{1}{y}dy=\int x dx$

and so:

$\ln |y|+c_1=\frac{x^2}{2}+c_2$

here we have one constant of integration from each side of the interval, but because they are just constants, we can put them into one constant and call it $c$:

$\ln |y|=\frac{x^2}{2}+c$

we can then rearrange this to give:

$|y|=e^{\frac{x^2}{2}+c}=e^ce^{\frac{x^2}{2}}$

We can then call $e^c$ just a constant, and let’s call it $y_0$ because we can see that when $x$ is zero, $|y|$ is just going to be given by this constant:

$|y|=y_0e^{\frac{x^2}{2}}$

This has two solutions (one where $y$ is positive, and one where it is negative), so we can choose one of them, depending on the initial condition for $y$.…

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## UCT MAM1000 lecture notes part 34 – differential equations part iii – Direction flows and Euler’s method

We haven’t yet studied any general ways to solve differential equations. In the first case of exponential growth we found an easy way to solve the equation, but for the logistic equation we just gave the solution and showed that it indeed satisfied the equation. Here we are going to look at some methods for finding not the exact solution, but approximations of the solutions. The first method is the method of Direction Fields and it will give us a good idea of what the solutions are going to look like. The second method, Euler’s method will give us an approximation to a single solution and we will be able to improve it to get arbitrarily good solutions to any differential equation (so long as there aren’t particularly nasty pathologies in the differential equation).

Direction Fields

Let’s take a differential equation:

$\frac{dy}{dx}=x+y$

Note that sometimes we will say explicitly that $y(x)$ and sometimes we will leave it implicit, because the equation has a derivative of $y$ with respect to $x$.…

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## UCT MAM1000 lecture notes part 32 – differential equations and rabbits moving at the speed of light

Up to now if I gave you an equation, and asked you to solve it for $x$ you would be, in general, looking for a value of $x$ which solved the equation. Given:

$x^2+3x+2=0$

You can solve this equation to find two values of $x$.

I could also give you an equation which linked $x$ and $y$ explicitly, and you could find a relationship between the two which, given a value of $x$ would give you a value of $y$. You’ve been doing this now for many years. Now we’re going to add a hugely powerful tool to our mathematical arsenal. We’re going to allow our equations to include information about gradients of the function…let’s see what this means…

We’re going to take everything that you learnt about integration and turn it into a way to model and understand the world around us. This is a very powerful statement and indeed differential equations are without a doubt the most powerful mathematical tool we have to understand the behaviour of everything from fundamental particles to populations, economies, weather, flow of wealth, heat, fluids, the motion of planets, the life of stars, the flight of an aircraft, the trajectory of a meteor, the way a pendulum swings, the way a ponytail swings (see paper on this here), the way fish move, the way algae grow, the way a neuron fires, the way a fire spreads…and so much more.…

## UCT MAM1000 lecture notes part x – summary

When you were a child you first learnt how to count. You learnt the relationship between a number and the quantity of objects around you. You quickly learnt to manipulate these numbers, you could add them, and you could multiply them, and you could subtract them. Subtracting them brought you to negative integers. Then you learnt how to divide them and this took you to fractions and thus the rational numbers. You could then manipulate these new numbers with the same operations.

At school you learnt about powers of numbers, and square roots of numbers, and you learnt about equations and how you could solve them to find the numbers which satisfied the equation. There were a few rules which acted like dead-ends however. You could never take the square root of a negative number. In fact any fractional power of a negative number should have filled you with trepidation (and possibly a frisson of excitement), though sometimes you saw that there were sneaky solutions to things like $(-1)^\frac{1}{3}$.…

## UCT MAM1000 lecture notes part 31 – complex numbers part ix

When we were playing around with partial fractions we appeared to make a bit of an assumption which was that the only forms that we had to deal with in the denominator of a fraction could always be written as a factor of either linear parts ($a+b x$) or quadratic parts which we could not factor into linear parts ($ax^2+bx+x$) where $b^2-4ac<0$, and of course multiple powers of these, for instance we could have terms like $(a+bx)^3$ in the denominator. How do we know that we can always split a polynomial up into these factors where the coefficients are real? Couldn’t it be for instance that if I gave you a cubic polynomial that all the roots were complex and so I couldn’t factor it in a way that every factor came out with real coefficients? It turns out that the answer is no, but we need a couple more ingredients to prove this.…

## UCT MAM1000 lectures notes part 30 – complex numbers part viii

So, we can do basic algebra with complex numbers, take powers of them, and apply trig and exponential functions to them. There isn’t much left that we might want to do, but taking powers of them is very important and also pretty easy. Let’s say we wanted to calculate the solution to the equation:

$z^2=-1$

Well, we know what $z$ is for this, because that’s what got us into this mess in the first place! We know that there are two solutions and they are $\pm i$. That is to say that if you multiply either of these numbers by themselves, you get -1, by definition. How about $z^2=i$. The first thing to do whenever you have to take the root of a complex number, or a real number (here the second root of $i$) is to convert that number into modulus argument form – it will be infinitely easier like that.…

## UCT MAM1000 lecture notes part 29 – complex numbers part vii

So, we know how to take the exponential of any complex number now. We do it by converting the exponential into the exponential of the real and imaginary parts separately, and then use the relationship between $e^{ia}$ and the $\cos$ and $\sin$ functions to write everything in terms of functions of real numbers, which we know how to deal with. How about the trigonometric functions applied to complex numbers? Well, we have a pretty good hint already from how we got from the exponential of complex numbers to trigonometric functions of real numbers. In fact we’re just going to give the answer, but you can work it out using Taylor series as well. For a complex number $z$:

$\cos z=\frac{e^{iz}+e^{-iz}}{2}$

$\sin z=\frac{e^{iz}-e^{-iz}}{2i}$

The first thing to check is that this is true when $z$ is a real number. It looks pretty strange at first site, especially the definition of $\sin$ because there’s an $i$ sticking out in the denominator like a sore thumb!…

## UCT MAM1000 lecture notes part 28 – complex numbers part vi

So last time we discovered that there was this amazing link between exponentials and trigonometric functions where the bridge between them was precisely complex numbers.

We now know how to take the exponential of any complex number and it is given by the exponential of a real number and a sum of terms which contain trig functions of real numbers:

$e^{a+ib}=e^a(\cos b+i\sin b)$

We can see also that now we have a third way to write a complex number. If you have a number in modulus argument form like:

$r(\cos\theta+i\sin\theta)$

Then we can also write this as $re^{i\theta}$. This is an alternative way of writing the modulus argument form.

In this form it also becomes more obvious that moduli multiply and arguments add. If we have two complex numbers:

$z=|z|e^{i\theta}$ and $w=|w|e^{i\phi}$ Then:

$zw=|z||w|e^{i(\theta+\phi)}$

The fact that we can now take the exponential of any complex number is very powerful. The point is that in order to calculate this function, all we need to be able to do is to take exponentials and trig functions of real numbers, and that we can do.…