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

[iFruit]

OK Lor:

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.

[atutor2001]

OK Lor:

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:

[mathqa]

iFruit:

OK Lor:

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/nonhomgeneous-differential-equations.html

MathQA

[OK Lor]

mathqa:

iFruit:

[quote=\"OK Lor\"]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/nonhomgeneous-differential-equations.html

MathQA[/quote]Hi,

Thanks to all the Maths Guru here

[iFruit]

mathqa:

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/nonhomgeneous-differential-equations.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.

[atutor2001]

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.

[mramk]

iFruit:

mathqa:

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/nonhomgeneous-differential-equations.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.

Thanks MathQA for volunteering your time to help solve this tough question.
Small mistake does not matter. It is just + or - of a constant value.
Overall steps posted on blog are well defined and easily followed.
But may I suggest you try to post the images of your works here. The forum does support [img] tag.

@iFruit: we need steps to understand how to solve the question, not just final answer. How to solve the problem is much more important than final answer. Appreciate if you could be more elaborate next time then :). Thanks!!!

Best regards,
Mr AMK.

[iFruit]

mramk:

@iFruit: we need steps to understand how to solve the question, not just final answer. How to solve the problem is much more important than final answer. Appreciate if you could be more elaborate next time then :). Thanks!!!

Best regards,
Mr AMK.

Hi Mr AMK,

Sure, I understand how to solve the problem is much more important than the final answer

I was just highlighting a procedural mistake in the solution ( not just + or - of a constant). My post was a conversation with MathQA, not my solution.

MathQA, I appreciate your time and effort too

Thanks.

[ChiefKiasu]

atutor2001:

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.

Honestly, I find differentiation and integration problems as \"routine\" and easily solvable once you master the methods and techniques. There are only so many different ways of solving the problems.
The P6 math questions tend to lean too much towards abstract problems to which kids cannot apply a \"standard\" technique. That is the difficulty of P6 math versus higher standard math.

[iFruit]

atutor2001:

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.

Hi atutor1,

Actually it is not so difficult as it sounds. It is a well defined procedure to solve the problem. That is why I was mentioning earlier the need to know the background theory.