So you have
where is a constant. Correct?
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.
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?
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.