Integral equations

• July 25th 2009, 05:07 AM
Twig
Integral equations
Hi!

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

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

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

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

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

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

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

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

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

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

Thanks!
• July 25th 2009, 05:21 AM
running-gag
Hi

It seems OK to me, I would have done the same

Just a typo here : $-e^{-y(x)}=-e^{-x}+C$
• July 25th 2009, 05:45 AM
Twig
Thank you