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May
29
answered If $f$ and $g$ are integrable functions on $I=[a,b]$ and if $h(x):=\inf(f(x), g(x))$ for all $x \in I$, prove that $h$ is integrable at $I$.
May
14
comment Bound on uniform norm of convolution of $L^p$ functions
Oh. I didn't even know that notation was still used!
May
13
comment Bound on uniform norm of convolution of $L^p$ functions
Don't you need $p^{-1}+q^{-1}=u^{-1}+1$?
May
13
answered Diffuse equation-type PDE: Help me!
May
12
comment How do I convert the limit definition of differentiability to different variables?
Because $h$ is just a displacement vector (from the point in question, $(x_{0},y_{0})$), and you send it to $0$. Being a vector, it has coordinates. Call them $(x,y)$, or anything you wish.
May
12
revised How do I convert the limit definition of differentiability to different variables?
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May
12
answered How do I convert the limit definition of differentiability to different variables?
May
12
revised A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$
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May
12
revised A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$
added 195 characters in body
May
12
answered A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$
May
12
answered Examine the convergence of $\left({\cos n\over n}\right)$
May
9
answered Show that $f \equiv 0$
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
See extended answer above.
May
9
revised Variation of parameters: $y''-y'-2y=4e^{-t}$
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May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
Well isn't that the question of the day; have you even attempted to compute it on your own?!
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
For example, $v_{1}$ (with my convention) is $-\int\frac{e^{2t}4e^{-t}}{3e^{t}}\;dt=-\frac{4}{3}\int\;dt=\frac{4}{3}t$.
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
Yes. And just proclaiming you can't do it doesn't help us to help you. I don't understand what's so difficult here; you're just computing integrals at this point, something you should have had lots of practice in before taking your current ODE course.
May
9
comment Questions about surface integrals and an example problem
What do you mean? It's just notation for integrals; you're not literally multiplying anything by $dS$. $\int_{S}(x+y)\;dS$ just means integrate $(x+y)$ over $S$ with respect to the surface measure $dS$. To reduce matters to an ordinary iterated Riemann integral, you parameterize the surface and pull back $S$ to the parameter domain $D$, and along the way $dS$ becomes $|\phi_{u}\times\phi_{v}|\;du\;dv$. This can all be made rigorous, and indeed, if you looked in your calculus text book you would find a sketch of the proof.
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
Your answer is fine. It depends on what you take to be $y_{1}$ and $y_{2}$; permuting them in the definition of $W$ will only change the sign, since $W$ is a determinant. But once you decide on your labeling, you must be consistant with the definitions for $v_{1}$ and $v_{2}$ (see the derivation of $v_{1}$ and $v_{2}$ in your text book).
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
$W[e^{-t},e^{2t}](t)=2e^{-t}e^{2t}-(-e^{-t}e^{2t})=2e^{t}+e^{t}=3e^{t}$.