Hi!

Problem:Find all continous solutions to $\displaystyle y(x)-2-\int_{0}^{x} \; e^{y(t)-t} \; dt =0$ and give their domains.

$\displaystyle y'(x)-e^{y(x)-x}=0 $ (, according to fundamental theorem of calculus)

$\displaystyle y'(x)-\frac{e^{y(x)}}{e^{x}} =0$

Can I separate the variables like $\displaystyle \frac{y'(x)}{e^{y(x)}}=\frac{1}{e^{x}} $ So now I would need to integrate both sides, right?

$\displaystyle \int \frac{1}{e^{y(x)}} \; dy = \int \frac{1}{e^{x}} \; dx $

$\displaystyle -e^{y(x)}=-e^{x}+C $

From here I solve for $\displaystyle y(x) $ , and use the fact that $\displaystyle y(0)=2 $

This gave me $\displaystyle y(x)=-ln(e^{-x}-1+e^{-2}) $ .

For the domain, is this $\displaystyle x \mbox{ such that } e^{-x}-1+e^{-2} >0 $ ?

That would give $\displaystyle x<-ln(1-e^{-2}) $

Kinda unsure about this though.

Thanks!