MathQA tutor - Ask your A-level Maths questions here!

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MathQA tutor - Ask your A-level Maths questions here!

Postby mathqa » Thu Oct 28, 2010 7:43 am

Hi all,

Allow me to open this humble thread to assist students who are preparing for their A-level Maths examination.

Within my ability and time budget, I will try to response to all questions.

For clarity, questions with geometry or complex formula, let's use PNG image format for posting.

MathQA Min Tan.

mathqa
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Postby OK Lor » Sun Oct 31, 2010 10:48 pm

Hi,

I'm not taking A-level Maths this year, but interested to know the solution for part (b):

If x(t) and y(t) are variables satisfying the differential equations
dy/dt + 2 dx/dt = 2x +5 and dy/dt – dx/dt = 2y + t,
(a) Show that 3 d²y/dt² - 6 dy/dt + 4y = 2 – 2t.
(b) Find the solution x in terms of t for the second order of differential equation given that y(0) = y’(0) = pi.

Thanks.

OK Lor
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Postby iFruit » Tue Nov 02, 2010 10:05 pm

OK Lor wrote:Hi,

I'm not taking A-level Maths this year, but interested to know the solution for part (b):

If x(t) and y(t) are variables satisfying the differential equations
dy/dt + 2 dx/dt = 2x +5 and dy/dt – dx/dt = 2y + t,
(a) Show that 3 d²y/dt² - 6 dy/dt + 4y = 2 – 2t.
(b) Find the solution x in terms of t for the second order of differential equation given that y(0) = y’(0) = pi.

Thanks.


Hi OK Lor,

Is this A-level question? AFAI can see, The equation is a non-homogeneous linear differential equation, so it needs some background to understand the solution..

iFruit
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Postby OK Lor » Tue Nov 02, 2010 10:39 pm

Hi iFruit,
This question is from Malaysia A level paper which I came across while browsing the web :!: .
Thanks.

OK Lor
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Postby iFruit » Tue Nov 02, 2010 10:53 pm

OK Lor wrote:Hi iFruit,
This question is from Malaysia A level paper which I came across while browsing the web :!: .
Thanks.


From what I understand ( the wording is not great), it is asking for a solution of the equation 3 d²y/dt² - 6 dy/dt + 4y = 2 – 2t given the initial conditions.

Let me know if you are interested, I can try and and work out a solution (need to recollect my linear diff equations first) but as I said it needs some background theory on linear diff equations first.

iFruit
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Postby atutor2001 » Wed Nov 03, 2010 1:04 am

OK Lor wrote:Hi,

I'm not taking A-level Maths this year, but interested to know the solution for part (b):

If x(t) and y(t) are variables satisfying the differential equations
dy/dt + 2 dx/dt = 2x +5 and dy/dt – dx/dt = 2y + t,
(a) Show that 3 d²y/dt² - 6 dy/dt + 4y = 2 – 2t.
(b) Find the solution x in terms of t for the second order of differential equation given that y(0) = y’(0) = pi.

Thanks.


diff both statement wrt to t
so the first one is y'' + 2x'' =2x'
second one is y'' - x'' =2y'+1

then simulataneous : statement 1 + 2 x statement 2
so we get 3y''=2x'+4y'+2

then we sub y'-x'=2y+t (2nd given eqn) into it

3y'' = 2(y'-2y-t) +4y' +2
3y'' = 2y'-4y-2t+4y'+2
3y'' = 6y'-4y-2t+2
3y''-6y'+4y =2-2t

Disclaimer : Above working not from me. Was chatting with my kid so took the opportunity to test :lol:

atutor2001
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Postby mathqa » Wed Nov 03, 2010 1:13 am

iFruit wrote:
OK Lor wrote:Hi,

I'm not taking A-level Maths this year, but interested to know the solution for part (b):

If x(t) and y(t) are variables satisfying the differential equations
dy/dt + 2 dx/dt = 2x +5 and dy/dt – dx/dt = 2y + t,
(a) Show that 3 d²y/dt² - 6 dy/dt + 4y = 2 – 2t.
(b) Find the solution x in terms of t for the second order of differential equation given that y(0) = y’(0) = pi.

Thanks.


Hi OK Lor,

Is this A-level question? AFAI can see, The equation is a non-homogeneous linear differential equation, so it needs some background to understand the solution..


iFruit is right to point out that this question is non-homogeneous linear differential equation. But it is nice to have for A level since it would test students's competence in many areas at once.

The solution is is posted at my blog. Cannot post it here due to oversize images.

http://mathqa.blogspot.com/2010/11/nonh ... tions.html

MathQA

mathqa
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Postby OK Lor » Wed Nov 03, 2010 12:03 pm

mathqa wrote:
iFruit wrote:
OK Lor wrote:Hi,

I'm not taking A-level Maths this year, but interested to know the solution for part (b):

If x(t) and y(t) are variables satisfying the differential equations
dy/dt + 2 dx/dt = 2x +5 and dy/dt – dx/dt = 2y + t,
(a) Show that 3 d²y/dt² - 6 dy/dt + 4y = 2 – 2t.
(b) Find the solution x in terms of t for the second order of differential equation given that y(0) = y’(0) = pi.

Thanks.


Hi OK Lor,

Is this A-level question? AFAI can see, The equation is a non-homogeneous linear differential equation, so it needs some background to understand the solution..


iFruit is right to point out that this question is non-homogeneous linear differential equation. But it is nice to have for A level since it would test students's competence in many areas at once.

The solution is is posted at my blog. Cannot post it here due to oversize images.

http://mathqa.blogspot.com/2010/11/nonh ... tions.html

MathQA


Hi,

Thanks to all the Maths Guru here :D

OK Lor
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Postby iFruit » Wed Nov 03, 2010 8:34 pm

mathqa wrote:
If x(t) and y(t) are variables satisfying the differential equations
dy/dt + 2 dx/dt = 2x +5 and dy/dt – dx/dt = 2y + t,
(a) Show that 3 d²y/dt² - 6 dy/dt + 4y = 2 – 2t.
(b) Find the solution x in terms of t for the second order of differential equation given that y(0) = y’(0) = pi.

The solution is is posted at my blog. Cannot post it here due to oversize images.

http://mathqa.blogspot.com/2010/11/nonh ... tions.html

MathQA


Hi MathQA,

Nice working. But I think there is a small mistake in it. You can't apply initial conditions to the homogeneous equation (equation 6) as they are initial conditions for non-homogeneous equation. That's why your solution doesn't tally back for initial conditions. Also particular solution C = -11/4

The solution after applying initial conditions to the general equation should be (pi = π)

x(t) = ((1-2π)/4) e^t cos(t/√3) - (√3(1+2π)/4) e^t sin(t/√3) - 11/4

Regards.

iFruit
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Postby atutor2001 » Wed Nov 03, 2010 9:48 pm

Whoa! So scary. Read so many times also no understand. I didn't know my poor kids have been subjected to such "torture". Must be nicer to them now.

So funny when comparing this to Pr Math. Parents are making so much noise when kids are in Pr but did not know what their older kids are facing.
Last edited by atutor2001 on Wed Nov 03, 2010 9:51 pm, edited 3 times in total.

atutor2001
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