Maths question [MTH1020]

[b^3]

Hah! I didn't even do spesh and I can do that

That's because from what I've seen the methods and spesh 'equivalent' units at uni actually cover a bit more, and in a bit more detail as well (which is good).

[alondouek]

haha I meant the blank stare part, I'm no good with the actual maths

[BigAl]

Sorry I didn't read the shell integration bit and assumed this was a volume of revolution question as seen in spesh

[lzxnl]

That's because from what I've seen the methods and spesh 'equivalent' units at uni actually cover a bit more, and in a bit more detail as well (which is good).

That shows how much more content VCE could and probably should cover. I mean, how long are the equivalent units at uni? A quarter to half the length?

[b^3]

That shows how much more content VCE could and probably should cover. I mean, how long are the equivalent units at uni? A quarter to half the length?

Off Topic

Normally a semester, so 12 weeks+exams. They don't look that much harder, but from looking at lectures notes from friends, they go through things more thoroughly.

I remember something I said last year, in relation to a few of the units (this is after spesh units). "It's 2-3 times the amount of content, twice as hard and done in half the time!" Although that may have something to do with engineering units trying to cram a few maths units into one, minu proofs"

Now we probably shouldn't take the thread off topic anymore.

Back to maths questions.

[alondouek]

Back to maths questions.

In that case, what's the best step-by-step procedure to solving differential equations? I have an assignment with a DE question due Thursday, and I don't even know what a DE is let alone how to solve one.

Help please?

[BigAl]

10 mins before I go to sleep

[lzxnl]

In that case, what's the best step-by-step procedure to solving differential equations? I have an assignment with a DE question due Thursday, and I don't even know what a DE is let alone how to solve one.

Help please?

Methods will seriously depend on the DE in question. There isn't a one-size-fits-all procedure for DEs. For example, y'=xy is easy to solve, but y'=x+y requires a bit more work and I'm not even sure how to solve y'=y^2+x. And these are just first-order ordinary differential equations. There are higher order DEs and partial DEs...which are horrible...

[alondouek]

So I need to 1. solve

and 2. "Find the unique function satisfying the following differential equation
with the given initial condition: ,

like, I have absolutely no idea what I'm looking at

[lzxnl]

So I need to 1. solve

and 2. "Find the unique function satisfying the following differential equation
with the given initial condition: ,

like, I have absolutely no idea what I'm looking at

OK these ones are part of a particular variety of DE called "separable equations".
Basically, you can rewrite these DEs as dy/dx = f(x)*g(y)

For the first one, we have 1/y*dy/dx = 4x/(1+x^2)
Integrating both sides, we get integral of dy/y = integral of 4x/(1+x^2) dx
I believe you can see pretty quickly that we get ln|ky|=2ln|1+x^2| where I've just put the arbitrary integration constant into the log.
Exponentiating both sides we get |ky|=(1+x^2)^2

For the second one, rewrite as dy/dx = x^2 y
dy/dx * 1/y = x^2
integral of dy/y = integral of x^2 dx
ln|ky|=1/3*x^3
|ky|=e^(1/3*x^3)

y=1 when x=1 so use that to find the particular solution.

The point is that you separate your DE into g(y) dy/dx = f(x)
Integrate both sides so integral of g(y) dy = integral of f(x) dx
and solve

[BigAl]

Alright, the basic idea is to collect dx and x in one side and y and dy in one side. So for the first question you can write it as (dy/y)=(4xdx/1+x^2) now you can integrate both sides and make y as a function of x. Don't forget +c
Second one is the same but you have to solve for c this time given initial values

[alondouek]

Thanks! I'll write this up and see if I can make heads or tails of it

...where I've just put the arbitrary integration constant into the log.

So is this arbitrary integration constant just the "+c" used with indefinite integrals?

Also, what's the point of these differential equations? What are they actually telling me? (I'm trying to understand the concept behind this as well as doing the question)

[BigAl]

Thanks! I'll write this up and see if I can make heads or tails of it

So is this arbitrary integration constant just the "+c" used with indefinite integrals?

Also, what's the point of these differential equations? What are they actually telling me? (I'm trying to understand the concept behind this as well as doing the question)

Yes k is the arbitrary constant. You can't always express something in the form a=cv+d sometimes you have to define relative rate of changes in one equation..I don't know what this equation is but it could be anything. You've probably heard of acceleration from F=ma. Now this is a differential equation and there are several ways you can express a such as vdv/dx

[lzxnl]

Thanks! I'll write this up and see if I can make heads or tails of it

So is this arbitrary integration constant just the "+c" used with indefinite integrals?

Also, what's the point of these differential equations? What are they actually telling me? (I'm trying to understand the concept behind this as well as doing the question)

The point? Sometimes physical models are given in terms of rates. An example is, of course, F=ma=m*d^2x/dt^2
So for a spring, where F=-kx, you have d^2 x/dt^2 = -kx/m

There are also spesh questions like you have a salt solution of a given volume and concentration, liquid coming in at v1 L/min containing concentration of the same salt c1, and the solution is well mixed and poured off at v2 L/min. That's another DE question.

[b^3]

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 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 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