Homework Statement
We have an inclined plane with a mass ##M## and an angle ##\alpha## and a box of mass ##m## over it.
Everything is at the instant 0 (it's a problem of static, no dynamics).
a) What's the acceleration in the component x of the box?
b) What's the acceleration in the component y...
My university library has almost no math books (I think it has 3 different books on differential equations and 1 on metric topology and that's the most advanced thing it has), but where I live there's an institute where they have engineering and physics undergrad and graduate students, it has a...
Sorry, I probably should have waited after reading Sturm - Liouville theory before approaching these exercises. I only had done exercises without reaching that chapter, but today I will read about this to be able to solve the problems in a more systematic way instead of having to try in this way.
Oh I think that I got it. I should use a series of cosines like
##
\cos ( (n+ \dfrac{1}{2} ) \dfrac{ \pi }{L} x )
##
In which case, it's the same as the series I used. I'm not sure how to do a reasoning to arrive this cosine series, but perhaps doubling the interval so I'm changing ## \dfrac{...
I may be understanding something bad, but:
$$
\int_{0}^{\pi} \cos (nx) dx = \dfrac{1}{n} \sin (nx) \bigg|_{0}^{\pi} = 0, \qquad \forall n \in \mathbb{N}
$$
I checked with calculator to see if I was making a mistake integrating and I still have this equal to zero, at least for n=1,2,3 and 4, but...
First of all, sorry for the poor redaction. I have re-read what I wrote and when I meant "as the zero eigenvalue is discarded" I wrote "as the constant eigenvalue is discarded" which makes no sense at all.
Yes, that's why I said that I have solution only for the constant equal to zero, I...
Homework Statement
I have to solve the following problem
$$
\left\{
\begin{array}{ll}
\dfrac{ \partial^{2} u }{ \partial x^{2} } + \dfrac{ \partial^{2} u }{ \partial y^{2} } =0 & \qquad \forall x \in (0, L), y > 0 \\
& \\
\dfrac{ \partial u }{ \partial x } (0,y) =0, & \qquad \forall y > 0 \\
&...
Oh. I'm self studying PDE, I have been given a book to follow but it has too few topics.
A good book to learn about green's functions? Is Evan's one recommended?
Laplace transform was very good, even for the case ##a= \omega ##
Working a little more I've got
$$
y(t) = \frac{1}{2} \dfrac{ b }{ \omega^{2} - a^{2} }
\bigg( a \sin ( ( \omega+a) x ) \sin (x \omega) + \omega \sin ( (\omega-a) x ) \cos (x \omega) \bigg) + c \sin (x \omega )
$$
I probably made a mistake in a previous step.
Homework Statement
I was reading a PDE book with a problem of resonance
$$
y_{tt} (x,t) = y_{xx} (x,t) + A \sin( \omega t)
$$
After some work it arrived to a problem of variation of parameters for each odd eigenvalue. To solve it, it uses
$$
y''(t)+a^{2} y(t) = b \sin ( \omega t) \qquad y(0)=0...
Yes, I may have phrases it wrongly. I will have to study Sturm Liouville Theorem. I know about basis from functional analysis course, what I tried to ask is why they failed to be a complete orthonormal set.
Oh, thank you very much. I supposed they could do it but wasn't really sure.
They are not dense of they're only not orthonormal?
I will check that myself.
I've solving some separation of variables exercises, and I have a doubt when it comes to the Laplacian
$$
u_{xx} +u_{yy} =0
$$
I usually have a rectangle as boundary conditions, so I use the principe of superposition and arrive to
$$
\dfrac{X''(x)}{X(x)} = - \dfrac{Y''(y)}{Y(y)} = - \lambda
$$...
Yeah, but I was trying to let this elementary. As they hadn't studied DE formally in this course, I didn't want to mention that's a DE.
But in the end what I'm doing is variation of parameters...
But yeah, your solution and huntman's one are way more straightforward, and even if I wanted to keep...