# Dominated Convergence & Differential Equation Limits

can a person kindly aid me with these couple of inquiries?

$\displaystyle \frac{dy}{dx} + \left( x+\frac1{x} \right) y = 1$ with $y(1)=0$.

I do not get what restrictions to absorb the integration variable and/or when incorporating both sides. I took care of the integration/method well - simply the restrictions are creating some troubles!

Allow $K(x,t)$ be a continual function specified for $x\in [a, +\infty)$ and also $t\in [c,d]$ where $a$, $c$, $d$ are dealt with actual numbers. What does it suggest to claim that $K(x,t)$ has controlled merging on $[a, +\infty) \times [c,d]$?

Locate $\displaystyle \frac{d}{dt} \left( \int_1^{+\infty} \frac 1x e^{-xt} dx \right)$ for $t>0$, showing plainly why your adjustments are warranted.

I took care of to do the integration (by taking the $\displaystyle \frac{d}{dt}$ inside the indispensable). I' m not exactly sure just how to warrant it, due to the fact that you ca n`t locate $\displaystyle \max \left(\frac1{x} e^{-xt}\right)$ using distinction.

p.s. this is test preparation not research!

Here are some tips, yet you can locate response to all 3 inquiries in any kind of calculus publication: -)

I do not get what restrictions to absorb the integration variable and/or when incorporating both sides.

Given that we are speaking about an average differential formula with suggested first value, I'm not fairly certain what you suggest by "limits to absorb the integration factor". If you are speaking about "separation of variables and also integration", after that the integration component has to do with locating the antiderivative of the integrant, not concerning doing a precise indispensable.

What does it suggest to claim that K (x, t) has actually controlled merging

I recognize the term "dominated convergence" from one context just, the Lesbegue controlled merging theory, see Wikipedia.

I took care of to do the integration (by taking the ddt inside the indispensable). I' m not exactly sure just how to warrant it

This is called the Leibniz indispensable regulation (Wikipedia), or even more usually distinction under the indispensable indicator. Every evaluation training course will certainly cover some theory that states enough problems such that the button of distinction and also integration stands, relying on the idea of indispensable (Cauchy, Riemann, Lesbegue indispensable