Help with exact equations

Hi all,

I'm having a really difficult time with this question:

t(1-x)-(t^2 * x)/2 * dx/dt = 0.

It has to be solved using the exact equations method, and via Young's Theorem, fx and gt are not the same:

fx = -x, gt= -tx

I'm struggling on actually finding a viable integrating factor, though! I've done the case (gt-fx)/g to get a function in terms of t only, but this leaves me with a really dodgy integrating factor, which when I try and solve the equation using, end up with some gamma function.

Any help, please?

Reply 1

ghostwalker

17

Original post by econhelp525

Hi all,

I'm having a really difficult time with this question:

t(1-x)-(t^2 * x)/2 * dx/dt = 0.

It has to be solved using the exact equations method, and via Young's Theorem, fx and gt are not the same:

fx = -x, gt= -tx

I'm struggling on actually finding a viable integrating factor, though! I've done the case (gt-fx)/g to get a function in terms of t only, but this leaves me with a really dodgy integrating factor, which when I try and solve the equation using, end up with some gamma function.

Any help, please?

Not my forté, but from a little digging and in lieu of any other responses, there are potentially two integrating fractors (and it's a case of checking each out) arising out of exponentating:

\displaystyle \int \frac{g_t-f_x}{g}\;dt

\displaystyle \int \frac{f_x-g_t}{f}\;dx

And the latter yields a very simple integrating factor.

Couldn't get LaTeX to produce elegant output when exponentiating both of those - anyone know how it might be done?

Reply 2

mqb2766

21

With a bit of rearranging you can do a fairly simple seperation of variables. You end up with something like
log(x) + x = log(t)
hence the problem with using elementary functions for writing down the closed form solution.

(edited 2 years ago)

Reply 3

econhelp525

OP

19

Original post by mqb2766

With a bit of rearranging you can do a fairly simple seperation of variables. You end up with something like
log(x) + x = log(t)
hence the problem with using elementary functions for writing down the closed form solution.

I've already solved it using that method, however we're asked to verify using the exact equations method. Thank you anyway

Reply 4

econhelp525

OP

19

Original post by ghostwalker

Not my forté, but from a little digging and in lieu of any other responses, there are potentially two integrating fractors (and it's a case of checking each out) arising out of exponentating:

\displaystyle \int \frac{g_t-f_x}{g}\;dt

\displaystyle \int \frac{f_x-g_t}{f}\;dx

And the latter yields a very simple integrating factor.

Couldn't get LaTeX to produce elegant output when exponentiating both of those - anyone know how it might be done?

The first one gives me an expression in terms of t only, which I tried using as an integrating factor, but got a horrible gamma function from. The second one is in terms of t and x, so not a valid integrating factor.

I just don't know what's going wrong here!

Reply 5

mqb2766

21

Original post by econhelp525

I've already solved it using that method, however we're asked to verify using the exact equations method. Thank you anyway

It helps to be clear in the OP what you've tried. When you say verify, do you mean verify that the solution satisfies the DE or try and redo the solution? Could you upload a pic of the original question and your working pls.

Reply 6

econhelp525

OP

19

Original post by mqb2766

It helps to be clear in the OP what you've tried. When you say verify, do you mean verify that the solution satisfies the DE or try and redo the solution? Could you upload a pic of the original question and your working pls.