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\[ \textbf{(Separable) Differential Equations} \\
\ \text{Evaluate } \int_{0}^{\infty} e^{t^2  (9/t^2)} \; dt \]
 one year ago
 one year ago
\[ \textbf{(Separable) Differential Equations} \\ \ \text{Evaluate } \int_{0}^{\infty} e^{t^2  (9/t^2)} \; dt \]
 one year ago
 one year ago

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AccessDeniedBest ResponseYou've already chosen the best response.0
\( \normalsize{ Hint \text{: Let} \\ \quad I(x) = \int_{0}^{\infty} e^{t^2  (x/t)^2} \; dt \text{.} \\ \qquad \color{red}{ \text{Calculate } I ' (x) \text{ and find a differential equation for } I(x) } \\ \qquad \text{Use the standard integral } \int_{0}^{\infty} e^{t^2} \; dt = \frac{\sqrt{\pi}}{2} \text{ to determine } I(0) \\ \qquad \text{Use this initial condition to solve for } I(x) \\ \qquad \text{Evaluate } I(3) \text{.}} \) I've been looking at this problem for a while, but I cannot figure out how to calculate that derivative. :P /first Q
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
But, I think I can handle the rest of the problem if I know how to calculate the I'(x)...
 one year ago

mukushlaBest ResponseYou've already chosen the best response.3
let\[ I(x)=\int_{0}^{\infty} f(x,t)\ \text{d}t\]\[I'(x)=\int_{0}^{\infty} \frac{\partial f}{\partial x} \text{d}t\]but there is some condition for this derivative
 one year ago

mukushlaBest ResponseYou've already chosen the best response.3
Let the Integral\[F(x)=\int_{c}^{\infty} f(x,t)\ \text{d}t\]be convergent when \(x \in [a,b]\) . let the partial derivative \(\frac{\partial f}{\partial x}\) be continuous in the 2 variables \(t,x\) when \(t>c\) and \(x \in [a,b]\) . and let the integral\[\int_{c}^{\infty} \frac{\partial f}{\partial x} \text{d}t\]converge uniformly on \([a,b]\). then \(F(x)\) has a derivative given by\[F'(x)=\int_{c}^{\infty} \frac{\partial f}{\partial x} \text{d}t\]
 one year ago

mukushlaBest ResponseYou've already chosen the best response.3
it was from my notes of the book : advanced calculus  taylor angus & wiley fayez
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
\[  \left (t^2 + {9 \over t^2}\right ) =  \left( t + {3 \over t}\right)^2 + 6\] Let, \[ \left( t + {3 \over t}\right) = u \\ du = \left( 1  {3 \over t^2} \right) dt\]
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
\[ t^2  ut + 3 = 0 \\ t = {u \pm \sqrt{u^2  12 }\over 2}\] this is ugly
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
let's try some brute method\[  \left (t^2 + {9 \over t^2}\right ) =  \left( t  {3 \over t}\right)^2  6\]
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
the limits of integration seems to change so badly t > inf, u>inf t> 0, u>inf
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
the problems remains the same ... i thought i would get rid of the denominator t^2
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
Hmm... Well, I tried using the partial derivative of the inside for x: \[ \int_{0}^{\infty} \neg \frac{2x}{t^2} e^{t^2  (x/t)^2} dt \] Which looks interesting, having the extra 1/t^2 involved now. I feel like maybe experiment has something there too, but I have to sleep for now. I'll go over this problem more tomorrow. Thanks for the help! :D
 one year ago

mukushlaBest ResponseYou've already chosen the best response.3
sure...Access this is beautiful !
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
Okay, I took some time earlier today and I think I finally got it. :D
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
\[ I(x) = \int_{0}^{\infty} e^{t^2  (x/t)^2} \; dt \\ \\ \begin{align} I'(x) &= \frac{d}{dx} \left( \int_{0}^{\infty} e^{t^2  (x/t)^2} \; dt \right) \\ &= \int_{0}^{\infty} \frac{\partial}{\partial x} \left( e^{t^2  (x/t)^2} \right) \; dt \quad (t > 0) \\ &= \int_{0}^{\infty} \left( \neg \frac{2x}{t^2} \right) e^{t^2  (x/t)^2} \; dt \\ &= \int_{0}^{\infty} \left( \neg \frac{2x}{t^2} \right) e^{(t  x/t)^2  2x} \; dt \\ &= \int_{0}^{\infty} \left( \neg \frac{2x}{t^2} \right) e^{2x} e^{(t  x/t)^2} \; dt \\ &= \neg 2e^{2x} \int_{0}^{\infty} \frac{x}{t^2} e^{(t  x/t)^2} \; dt \\ & \quad u = t  \frac{x}{t} \\ & \quad du = 1 + \frac{x}{t^2} \; dt \quad \text{Add/subtract to get the +1 coefficient} \\ & \qquad \implies \text{Upper Bound: } \lim_{t \to \infty} u = \infty \text{,} \\ & \qquad \implies \text{Lower Bound: } \lim_{t \to 0^{+}} u =  \infty \text{. (From:} t>0) \\ &= \neg 2e^{2x} \int_{0}^{\infty} \left( \frac{x}{t^2} e^{(t  x/t)^2} + e^{(t  x/t)^2}  e^{(t  x/t)^2} \right) \; dt \\ &= \neg 2e^{2x} \int_{0}^{\infty} \left(1 + \frac{x}{t^2} \right) e^{(t  x/t)^2}  e^{(t  x/t)^2} \; dt \\ &= \neg 2e^{2x} \left( \int_{0}^{\infty} \left(1 + \frac{x}{t^2} \right) e^{(t  x/t)^2} \; dt  \int_{0}^{\infty} e^{(t  x/t)^2} \; dt \right) \\ &= \neg 2e^{2x} \int_{0}^{\infty} \left(1 + \frac{x}{t^2} \right) e^{(t  x/t)^2} \; dt + 2e^{2x} \int_{0}^{\infty} e^{(t  x/t)^2} \; dt \\ &= \neg 2e^{2x} \int_{\infty}^{\infty} e^{u^2} \; du + 2 \int_{0}^{\infty} e^{t^2  (x/t)^2} \; dt \\ I'(x) &= \neg 2e^{2x} \sqrt{\pi} + 2 I(x) \end{align} \] It 'seems' nice.. :P
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
Although, the resulting equation does not seem to be a separable DE... it looks more like a linear DE.
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
Yep .. linear in x
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
one way to do it \[ Let,  \left (t^2 + {9 \over t^2}\right ) =  \left( t  {3 \over t}\right)^2  6 \\ \] you havedw:1346277919237:dw
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
you have to prove that \[ \int_0^\infty e^{(x  a/x)^2} dx = \int_0^{\infty} e^{x^2} dx= {\sqrt \pi \over 2 }\]
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
I think this is pretty much solved\[ \int_{0}^{\infty} \left(1 + \frac{x}{t^2} \right) e^{(t  x/t)^2} \; dt \]
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
this is particularly interesting. I didn't know it before. \[ \int_0^\infty e^{(x  a/x)^2} dx\]
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
Hmm, this has been a very interesting problem for me. I never considered using a perfect square there at the beginning, but it really turned out nicely. Plus, this standard integral stuff is always cool, with that square roots of pi coming up. Thanks for all the help! :D
 one year ago

mukushlaBest ResponseYou've already chosen the best response.3
allow me to do some effort here.\[\text{I}(x)=\int_{0}^{\infty}e^{t^2(\frac{x}{t})^2} \ \text{d}t\]\[\text{I}'(x)=\int_{0}^{\infty}\frac{2x}{t^2}e^{t^2(\frac{x}{t})^2} \ \text{d}t\]let \(u=\frac{x}{t}\) \(x>0\)\[\text{I}'(x)=\int_{\infty}^{0}2e^{(\frac{x}{u})^2u^2} \ \text{d}u=2\int_{0}^{\infty }e^{(\frac{x}{u})^2u^2} \ \text{d}u=2I(x)\]\[\text{I} '(x)+2\text{I}(x)=0\]\[\text{I}(x)=ce^{2x}\]\[c=\text{I}(0)=\frac{\sqrt{\pi}}{2}\]\[\text{I}(x)=\frac{\sqrt{\pi}}{2e^{2x}}\]
 one year ago

experimentXBest ResponseYou've already chosen the best response.1
yeah ... the trick was y=x/t ... this could have been done without DE. I like the concept :)
 one year ago
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