Thread: Economics Problem

So I've never taken differential equations, but this problem popped up in an Econ class so I'm pretty lost.

Let R(w) = {(-w)[u''(w)]}/u'(w)

Suppose R(w) = r > 0 for all w.

Derive an expression for u(w).

So i began by multiplying both sides by u'(w) and then bringing everything over to the left. I'm not sure where to introduce d/dw. I guess I'm pretty lost. There was a hint: let z(w) = u'(w) and solve the differential equation for z.

This then becomes:
R(w) = {(-w)[z'(w)]}/z(w)
[z(w)]R(w) = -w[z'(w)]
[z(w)]R(w) + w[z'(w)] = 0

Then I'm stuck.
Thanks in advance for the help.

So you have

where is a constant. Correct?

correct!!

Right then. I wouldn't use the R(w) notation, because that's essentially an extra variable. Follow the hint (which reduces the order of the DE by one), and you get

You can divide both sides by to obtain



You can now integrate both sides directly. You obtain

Can you continue from here?

do i also have to multiply each side by d/dw ?

No. Right now, I would solve for z(w), and then plug back in what that was. What do you get?

z(w) = e ^ -r ln(w)

u(w) = the above integrated

i wish i knew how to type this math notation...

You can simplify the above expression a good deal:



Incidentally, I forgot a constant of integration. We would get that back at this step:

which would translate to the multiplication of an arbitrary constant (equal to ), which gives us

Now continue?

Incidentally, you can learn something about LaTeX here.