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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE General & Further Mathematics => Topic started by: sam.utute on October 19, 2010, 06:27:41 pm

Title: Solving Difference Equations
Post by: sam.utute on October 19, 2010, 06:27:41 pm: Unfortunately, the MathsQuest textbook that my school uses has absolutely nothing about this topic in it, so I've been learning about from making mistakes and checking the suggested solutions. Is this topic an assessable part of the study design? And can someone please explain it to me.

Please explain how to solve this equation (incidentally, it is from the Neap study guide)

S(n) = 1.06 x S(n-1) - 300, S(1) = 39000

Also, I have seen two different equations to solve the generic difference equation. Which one is right?

t(n+1) = at(n) + b, t(1)

Solution 1:
t(n) = a^(n-1) x t(1) + b x (a^(n-1) - 1)/(a-1)
Solution 2:
t(n) = a^(n) x t(1) + b x (a^(n) - 1)/(a-1)

Thanks :),
Sam
Title: Re: Solving Difference Equations
Post by: TrueTears on October 19, 2010, 06:45:08 pm: Use iteration

$s_n = 1.06 \times s_{n-1}$ where $s_1 = 39000$

$s_{n-1} = 1.06s_{n-2}$

$s_n = 1.06^2s_{n-2}$

$s_n = 1.06^ks_{n-k}$

Let $k = n-1$

Thus $s_n = 1.06^{n-1}s_1$

$s_n = 1.06^{n-1}(39000)$
Title: Re: Solving Difference Equations
Post by: ChairmanMao on October 19, 2010, 08:25:36 pm: Quote from: TrueTears on October 19, 2010, 06:45:08 pm
Use iteration

$s_n = 1.06 \times s_{n-1}$ where $s_1 = 39000$

$s_{n-1} = 1.06s_{n-2}$

$s_n = 1.06^2s_{n-2}$

$s_n = 1.06^ks_{n-k}$

Let $k = n-1$

Thus $s_n = 1.06^{n-1}s_1$

$s_n = 1.06^{n-1}(39000)$

.. Wow.
Title: Re: Solving Difference Equations
Post by: kamil9876 on October 19, 2010, 08:41:54 pm: actually TT u missed the -300.

$s_n=1.06s_{n-1} -300$

but yeah it isn't that hard to fix your solution to account for that. Anyhow if you do the general case you would just get:

Quote
Solution 1:
t(n) = a^(n-1) x t(1) + b x (a^(n-1) - 1)/(a-1)
Title: Re: Solving Difference Equations
Post by: sam.utute on October 19, 2010, 08:54:02 pm: Quote from: ChairmanMao on October 19, 2010, 08:25:36 pm
Quote from: TrueTears on October 19, 2010, 06:45:08 pm
Use iteration

$s_n = 1.06 \times s_{n-1}$ where $s_1 = 39000$

$s_{n-1} = 1.06s_{n-2}$

$s_n = 1.06^2s_{n-2}$

$s_n = 1.06^ks_{n-k}$

Let $k = n-1$

Thus $s_n = 1.06^{n-1}s_1$

$s_n = 1.06^{n-1}(39000)$

.. Wow.

Agreed. Wow...
So whenever I get a question that asks me to solve a difference equation, just substitute into the forumla?
Title: Re: Solving Difference Equations
Post by: TrueTears on October 19, 2010, 09:37:04 pm: Quote from: kamil9876 on October 19, 2010, 08:41:54 pm
actually TT u missed the -300.

$s_n=1.06s_{n-1} -300$

but yeah it isn't that hard to fix your solution to account for that. Anyhow if you do the general case you would just get:

Quote
Solution 2:
t(n) = a^(n) x t(1) + b x (a^(n) - 1)/(a-1)

lol opps didn't see it, anyways same shit just add the -300 at the end :D
Title: Re: Solving Difference Equations
Post by: kamil9876 on October 19, 2010, 10:01:47 pm: no not exactly:

it would go like this:

$s_n=1.06s_{n-1} -300$

$=1.06(1.06s_{n-2} - 300) -300$

$=1.06(1.06(1.06s_{n-3} -300) - 300 ) - 300$

etc.

Then when you expand everything out you get something like:

$<br />s_n=1.06^{n-1}s_1 + -300 + 1.06(-300) + 1.06^2(-300) ... + 1.06^{n-2}(-300)$

And then using the formula for the geometric sum you get the formula the OP provided.
Title: Re: Solving Difference Equations
Post by: TrueTears on October 19, 2010, 10:22:56 pm: yeah... which is why i said just add the -300 at the end and it's the same, i don't see any different logic in your working, your method was exactly the same as mine except when you iterated you added a -300 at the end lol..
Title: Re: Solving Difference Equations
Post by: kamil9876 on October 19, 2010, 10:25:31 pm: o ok i thought by 'same shit just add -300' u meant $s_n=1.06^{n-1} - 300$ (ie add it to the final answer).
Title: Re: Solving Difference Equations
Post by: TrueTears on October 19, 2010, 10:31:36 pm: haha yeah true, but yeah u get the point :P