Differential Equation with an Interval

[Orphen89]

Homework Statement

(1 + 2*x exp(−y))dx − (1 + x^2) exp(−y)dy = 0 with y(0) = 0

Show that the equation is exact and hence find the solution of the initial value problem on the interval 0 ≤ x < 1.

The Attempt at a Solution

For the differential equation, after solving everything with the initial value problem, I got the equation:

u(x,y) = x + x^2 * exp(-y) + exp(-y) = 1

(I don't really want to type out all the working out because it would take forever...)

Now, I'm not sure if this is correct, because it doesn't seem to fit over the interval given above. But then again, I'm not completely sure what this question is asking (yup, I'm pretty stupid). Can someone please check if what I have his correct?

Thanks in advance.

 

[Dick]

How do you mean 'it doesn't seem to fit over the interval'? It looks fine to me.

 

[Orphen89]

How do you mean 'it doesn't seem to fit over the interval'? It looks fine to me.

Ah okay, I think I know what you mean now - I was taking the value for c = 1 as the interval over 0 ≤ x < 1, so I thought I got it wrong.

Also, thanks for all the help Dick you've given me this week (you actually helped me out with a PMF question earlier this week as well which I didn't get to thank you for =/ )