# Personal Wealth Model using DE's

[SOLVED] Personal Wealth Model using DE's

Hey, I am having trouble with this question.

Most people have an income that comes from 2 sources: salary and personal investments. From this income, 'necessary' expenses (housing, food) are paid, some money is spent on luxuries and the rest is saved (increasing investments).

Given that income must equal outflow, show the steps in developing the following mathematics model for a person's wealth at any time t:

dW/dt=(1-p)(s-n+rW)
W(t)= your wealth (savings), which is a function of time
r= rate of interest on your wealth (savings)
n= amount spent on necessities
p=proportion of your income after necessities that you spend on luxuries.

Ok what I have to try to figure out is how to differentiate this model (if possible) to get it to the form of:

W(t)=((s-n)/r)(e^((1-p)rt)-1)

I'm not sure whether this is sufficient information as I left what appeared to be a lot of irrelevant information out.

Thanks very much for your help.

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Hootenanny
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Firstly, I'm not quite sure what the question is asking. Are you given this,
dW/dt=(1-p)(s-n+rW)
Or do you have to derive it?

We are given that function and are then asked to differentiate it (if possible), and if we can it should equal the second model for W(t), which was also given to us.

In other words we have to show the steps to obtaining the W(t) function from the original function.

Sorry if this isn't making much sense, if you want me to explain futher I will.

Hootenanny
Staff Emeritus
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We are given that function and are then asked to differentiate it (if possible), and if we can it should equal the second model for W(t), which was also given to us.

In other words we have to show the steps to obtaining the W(t) function from the original function.

Sorry if this isn't making much sense, if you want me to explain futher I will.
No, it makes perfect sense. However, there's one thing that we have to fix up before we can start. The equation which you quoted,
dW/dt=(1-p)(s-n+rW)
Is a differential equation for w(t) and therefore we need to integrate it to find an expression for w(t) not differentiate it.

So what type of differential equation is it?

Seems like there is not much to derive if all of that is part of the givens. I'm guessing s, r, n, p are assumed to be constant functions so basically you have a DE thats first order and linear in W, so use separation of variables and you'll find W.

No, it makes perfect sense. However, there's one thing that we have to fix up before we can start. The equation which you quoted,

Is a differential equation for w(t) and therefore we need to integrate it to find an expression for w(t) not differentiate it.

So what type of differential equation is it?

Yes I realised that I need to integrate it last night before I went to sleep.

I am a bit confused by your question... Do you mean what method should I use to solve it? Like partial fractions or DE's with separable variables? If so I think I should be using partial fractions.

Seems like there is not much to derive if all of that is part of the givens. I'm guessing s, r, n, p are assumed to be constant functions so basically you have a DE thats first order and linear in W, so use separation of variables and you'll find W.
ok I think I understand that. The thing thats confusing me is when I have to integrate it, there is no 't' to integrate with respect to. Does this matter at all?

Hootenanny
Staff Emeritus
Gold Member
I am a bit confused by your question... Do you mean what method should I use to solve it? Like partial fractions or DE's with separable variables? If so I think I should be using partial fractions.
Sounds good to me . However, there's no need to use partial fractions,

$$\frac{dW}{dt} = (1-p)(s-n+rW) \Rightarrow \frac{1}{s-nr+W} \frac{dW}{dt} =1-p$$

For $s\neq nr+W$. Notice that everything is constant except for W. Can you take it from here?

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umm I have to go to school now, I will give it a shot there and let you know.

Thanks

P.S. should it be s-n+rW?

Hootenanny
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P.S. should it be s-n+rW?
Of course it should, sorry it was a typo, which I corrected appropriately above.

I tried to intgrate it by taking dt over to the other side and integrating both sides. But I am not sure how to go about integrating it. For example on the right side does it become t+pt + c, and I don't have a clue how to integrate the left side.

Hootenanny
Staff Emeritus
Gold Member
I tried to intgrate it by taking dt over to the other side and integrating both sides. But I am not sure how to go about integrating it. For example on the right side does it become t+pt + c, and I don't have a clue how to integrate the left side.
You're almost correct, just watch your signs. Now for the LHS we have,

$$\int\frac{dW}{s-nr+W}$$

Notice that the denominator is a linear function of W, how do you usually integrate one over a linear function?

HINT:

$$\frac{d}{dW}\left(s-nr+W\right) = 1$$

Ummm do you take out the coefficient of what you're integrating with respect to, in this case it would be 1.

then it would be 1/1 ln(s-nr+W)?

Also, why does the r move off the W and onto the n? I don't quite get that sorry.

Hootenanny
Staff Emeritus
Gold Member
Ummm do you take out the coefficient of what you're integrating with respect to, in this case it would be 1.

then it would be 1/1 ln(s-nr+W)?

Also, why does the r move off the W and onto the n? I don't quite get that sorry.
Sorry, I've been making a lot of typos recently. The r should remain the coefficient of the W.

no that's quite allright. I appreciate your help.

So was my method of integrating correct?

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Hootenanny
Staff Emeritus
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no that's quite allright. I appreciate your help.

So was my method of integrating correct?
Yes it was, but note that your integral will change since W now has a coefficient r.

ok so it would be

1/r ln (s-n+W)=t-pt+c
ln (s-n+W)=tr-ptr+c
s-n+W=e^((1-p)rt)*e^c
W=(A*e^((1-p)rt))-s+n, where A=e^c

?

Hootenanny
Staff Emeritus
Gold Member
You seem to be missing the coefficient of W in your logarithm I'm not sure I understand what you mean. I don't really understand what I'm doing above, I am just going through rules we learnt in class.

Are you trying to say that when i integrate it should be 1/r ln (s-n+rW)?

Defennder
Homework Helper
Yes, that should be correct. I got the same answer.

Hootenanny
Staff Emeritus
Gold Member
1/r ln (s-n+W)=t-pt+c
Are you trying to say that when i integrate it should be 1/r ln (s-n+rW)?
The latter solution looks good to me Notice the factor of r in your second answer.

ok thanks, so is it possible to get it to look like the wealth function W(t). I guess before I do that I would have to find c. Not sure how to do it though.

Defennder
Homework Helper
You need an initial condition before you can find C, the constant of integration. This is given explicitly in the question, but you can think about it. What is the value of W at t=0?

Once you figure that out you can obtain the equation as given by your first post.

ok thanks. I would assume that when t=0, W=0.

I just dont understand one thiing. In the wealth model it appears to be ((s-n)/r)*(e^((1-p)rt)-1).

Currently, the s,n and r are just either added or subtracted.

Defennder
Homework Helper
Huh? What does that mean? Don't you get the answer when you assume W(0)=0? By the way I meant "not given explicitly" but i couldn't edit my post.