can someone kindly help me with these few questions? :)
$\displaystyle \frac{dy}{dx} + \left( x+\frac1{x} \right) y = 1$ with $y(1)=0$.
I do not get what limits to take in the integration factor and/or when integrating both sides. I managed the integration/method well - just the limits are causing some problems!
Let $K(x,t)$ be a continuous function defined for $x\in [a, +\infty)$ and $t\in [c,d]$ where $a$, $c$, $d$ are fixed real numbers. What does it mean to say that $K(x,t)$ has dominated convergence on $[a, +\infty) \times [c,d]$?
Find $\displaystyle \frac{d}{dt} \left( \int_1^{+\infty} \frac 1x e^{-xt} dx \right)$ for $t>0$, indicating clearly why your manipulations are justified.
I managed to do the integration (by taking the $\displaystyle \frac{d}{dt}$ inside the integral). I’m not sure how to justify it, because you can’t find $\displaystyle \max \left(\frac1{x} e^{-xt}\right)$ via differentiation.
p.s. this is exam prep not homework! :)