Maths question [MTH1020]

[alondouek]

As nlui has said, mostly for physical models, when you have one thing changing respect to another thing, or even a few variables changing with respect to a few other variables.

There's a few things to do with heat transfer and such, probably above the level of what you're learning now but
which has a general solution .

i.e. Given initial conditions and boundary conditions you can tell how the temperature along the 1-D rod varies in time and space. This can be extended to 2 and 3 dimensions.

Or for example a differential equation governing freefall.
The diff equation you have is , which has a partly ugly solution, .

There's a lot of applications in engineering for differential equations, even electrical circuits and such.

So yeah, really anytime there is a relationship of something changing with respect to something else changing really.

Thanks for this explanation, really puts the concept into context! If DEs measure change between variables, I wonder if they can be used to model population growth and decline in a biological setting?


Also, when I'm writing Riemann sums, I use notation such as . What exactly does this mean (I mean, I know sigma means "the sum of", but what are the above and below parts telling me)?

[b^3]

I couldn't get the edit in quick enough

There's stuff to do with springs and damping, i.e. you have a mass on a spring, and a damper and a force which is applied to the system, i.e. a 'forcing function'. In different situations the system will act differently, i.e. it could be underdamped, meaning it keeps oscillating but the oscillations eventually die down to zero. It could be critically damped at which the mass returns to the equilibrium position in the shortest possible time, or even overdamped, at which the damping force takes longer to return to the original position. Then with the forcing function, if the frequency of oscillation is correct, relative to the natural frequency then you could have the system going off to resonance, at which things like this happen: http://www.youtube.com/watch?v=j-zczJXSxnw

I guess there's also stuff in life sciences, I just don't know as much about it. Like there's stuff governing growth rates of populations and radioactive decay, and there's a lot to do with fluid flow and such as well, (I've heard about some life sciences students doing stuff with fluid flow through blood vessels and such. It annoys me that MBBS kids do stuff and don't denote partial derivatives correctly! arrrg)

There's a lot of applications in engineering for differential equations, even electrical circuits and such. So normally with Riemann sums you're relating it t integration, so you're summing up rectangles under the curve, that you're using to approximate the curve. You've split the curve into rectangles as I'm assuming you had points, and the is just an index.

So yeah, really anytime there is a relationship of something changing with respect to something else changing really.

EDIT: Added a few more examples, ahh procrastination

Basically you're summing up whatever is 'inside' the sigma from to (which for Riemann sums normally means rectangles approximating the curve). I have a feeling it should be , not though, but basically you're summing up those rectangles to get the area under the curve.

EDIT: I probably should add, since I was doing engineering maths at the time that we would have covered Riemann sums, I never really actually covered them at uni, so yeah the above is just based on extra reading and such. Correct me if I've said anything incorrect. Although the concept is kinda hidden away inside the VCE course.

[BigAl]

One of the examples the lecturer gave us last semester was relative population growth of foxes and rabbits...while one is decreasing exponentially, other one is increasing...so you may have two rate of change in one equation...to solve such equation, you typically need one more just like solving simulteneous equations

[alondouek]

So for the question , I did:

















Am I even close? 

[b^3]

Firstly, is it meant to be partial derivatives? It looks like it should just be the single derivative, i.e. not , since we only have a function of one variable.

Secondly, when you integrate it, you need a or or whatever on the line with your integrals. So you want


Thirdly, since we're not restricting the domain, you need to keep the modulus when you bring the log in, it can be removed later because of the constant though, as long as you define , , since the then will account for the when you undo the modulus. As you have done, we won't need a mod on the right hand log since the inside of the log is always positive.

[alondouek]

Yep, I've got the integrals written down as you have them haha, I'm just trying to learn Latex (for what it's worth). Definitely meant to be single derivatives.

If I'm bringing in the modulus for the left-hand side of the equation, should it be ?

Also, what exactly is ? Is it just a constant? (I'm really, really iffy on the theory for DEs)

[b^3]

.

Yeah is just a constant. Hence the .

EDIT: Also being a little bit more picky (this will depend on your lecturer though). Don't put therefore, on every line. Apparently it's more for when you're finishing with a result, i.e. the last line at the end. You can have an implies, though for the earlier lines, if you really want to though.

Although with that being said, I'm sure if we're being really picky that part of what I've just added isn't right as well

[alondouek]

Thanks! Does that mean I can express my final answer as ?

[BigAl]

Thanks! Does that mean I can express my final answer as ?

Yep make sure you specify the domain of A ie A belongs to R...some people are really picky about this
Edit

[BigAl]

Wait did you make y subject...i mean did you raise both sides by e?

[alondouek]

Wait did you make y subject...i mean did you raise both sided by e?

Sure did!

-->

[BigAl]

Im not feeling really well now..cant follow the steps..but im sure youve done it correctly

[BigAl]

Sorry my bad..but i think you need to have e after A

[alondouek]

Okay, spent half an hour on the phone to a friend who tried to explain the question to me from a theory standpoint, this is what I ended up with:



Is this better?

[alondouek]

Also, in the case where I need to find a unique function for ,

would be a valid answer?