# Dominated Convergence & Differential Equation Limits

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! :)

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Please edit your questions, or learn latex. I edited your previous questions also. – user9413 May 21 '11 at 18:55
And ask one question per post. – user9413 May 21 '11 at 18:56
and more importantly try out your homework problems before posting them here. Your homework is for you to gain and not for us to gain – user17762 May 21 '11 at 18:58
You have posted 4 questions in the span of half an hour, which makes me say that You haven't tried hard – user9413 May 21 '11 at 19:00

Here are some hints, but you can find answers to all three questions in any calculus book :-)

I do not get what limits to take in the integration factor and/or when integrating both sides.

Since we are talking about an ordinary differential equation with prescribed initial value, I'm not quite sure what you mean by "limits to take in the integration factor". If you are talking about "separation of variables and integration", then the integration part is about finding the antiderivative of the integrant, not about doing a definite integral.

What does it mean to say that K(x,t) has dominated convergence...

I know the term "dominated convergence" from one context only, the Lesbegue dominated convergence theorem, see Wikipedia.

I managed to do the integration (by taking the ddt inside the integral). I’m not sure how to justify it...

This is called the Leibniz integral rule (Wikipedia), or more generally differentiation under the integral sign. Every analysis course will cover some theorem that states sufficient conditions such that the switch of differentiation and integration is valid, depending on the notion of integral (Cauchy, Riemann, Lesbegue integral...). You simply need to look up which version was used in your class and therefore which conditions you need to prove (or at least tell us in your question, so that we can help you with that).

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Differential question question. When you work out the integration factor its e^ integral (x+ 1/x) right? But what are you integrating against ie. the limits? Thats what im confused about. Also its a first orrder DE so when you have d/dt(yIF) and then bring the dt over (ie integrate both sides) whats the limit on the integral? 0 to t? – user4645 May 22 '11 at 13:06
The definition of dominated convergence we use is mod(K(t,x)) <= p(t) where integral p(x) converges. I get what the definition is trying to say but im not always sure how to get that p(x) function! – user4645 May 22 '11 at 13:07
ODE: The integral in your formula represents the antiderivative, so there are no integration limits. – Tim van Beek May 22 '11 at 18:30
Dominated convergence: Sorry, I still don't understand what you are asking. Let's say we have a sequence of integrable functions $g_n(x)$ converging pointwise, and would like to know if the limit of the integrals is equal to the integral of the limit. If we find an integrable function $f(x)$ that dominates all the $g_n(x)$, then the answer is 'yes' according to the theorem of dominated convergence. First of all, I think you should clarify which variable it is you integrate over (x? t? (x, t)?). – Tim van Beek May 22 '11 at 18:35
Say you take the derivative within the integral. im not sure how to prove that step is valid. Thats about it. Hence the example above. Btw - i got it on the differential equations limits thing so you can ignore that. – user4645 May 24 '11 at 10:51