
Integral equations
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(1e^{2}) $
Kinda unsure about this though.
Thanks!

Hi
It seems OK to me, I would have done the same
Just a typo here : $\displaystyle e^{y(x)}=e^{x}+C $
