Laplace's Equation Puns

I'm taking an E&M class and using Griffith's *Electrodynamics* textbook and in the section about Laplace's equation he talks about it like the it's most amazing thing ever, but I don't get it. Why do we care so much about the case when the charge density is zero? If I understand the Laplacian operator correctly, that would mean the E field has to have a constant value in that region (although it feels like it should be zero, a physicist friend assures me that's not necessarily the case) and that doesn't sound very interesting. I mean, I get why the equation would be important to other branches of physics since the Laplacian being zero in vector calculus is analogous to the first derivative being constant in single variable calculus and those kinds of functions are just easier to work with, but I don't get why it's so important in E&M in particular.

Hey, can anyone help me in solving this kind of problems?

This needs to be solved using the Laplace transform.

(I know how to solve y''+y'+y=0 or even non-homogenous, this just bugs me.)

So I've been trying to help a friend with a laplace transform equation, and for the life of me I can begin to understand how to solve the last segment in this problem. There's two functions for the transform, so the multiplication rule applies right? for lack of being able to send special characters this is the equation, I can also send an image of what we have solved so far if needed. https://imgur.com/a/XarEdMh
Please help!
Thanks in advance!

https://preview.redd.it/97di3l5s5hn71.png?width=1101&format=png&auto=webp&s=acb28a2e5ad862c00cd537d58557bc5b75e651ad

Hello,

The following initial value problem is to be solved using methods of Laplace transforms, however it's forcing term is a Heaviside Function (Unit Step FUnctions)

I know I will have to take the laplace of both sides of the given equation dx/dt.

Step 1 i thought was to re-write the given equation accounrding to t H(t)=h, 0<=t<=Lambda H(t)=h, 1<=t<=(1+lambda) 0 Otherwise

We know that H(t) is the heat generated from people machines etc and is always positive (increaing) [0,infinity)

Is my first step to find x(t) by using the initial conditions from 0<=t<=(7/4)hr in order to get a general solution to the DE. After substituting, I would solve it using laplace transforms

Since x(0)=T* then at 0<=t<=(7/4), x(0)=18

I was able to find that x(t)=18 ? Please verify.

If that is indeed the first step how do I incorporate the use of the unit step fuction/ window functions. When do i express in unit step functions or do it?

Also, should i be evaluating using laplace the DE as giving (without putting in initial conditions)

Please help and thank you so much in advanced!

https://preview.redd.it/l9gl2xmq5hn71.jpg?width=2248&format=pjpg&auto=webp&s=c6bb49be2a4d5ff67c8f7917dc8ca208fadb15cc

https://preview.redd.it/6g6c6wmq5hn71.jpg?width=2216&format=pjpg&auto=webp&s=16c808e84b32ca434c77fcad74204529e3bb4394

https://preview.redd.it/dg7v5ymq5hn71.jpg?width=2064&format=pjpg&auto=webp&s=5c8d6829c2b8b69afffed71d7d2157b60270e5ee

Just finished Diff Eq and professor didn't seem to have time to finish his syllabus. We finished with higher order linear ODEs and completely left out laplace and systems if equations which I know is pretty important to learn. I could probably learn it on my own but just wanted to know if this is legit as an EE student or did I just waste my time in this class? Thoughts?

I ask for help with this f(t) https://i.stack.imgur.com/FLkD2.jpg The 1 is the heaviside function I used the timeshifting property to get this https://imgur.com/a/V0S38OA Is it the correct way? What should i do next with the sum?

Recent ME grad here working in the power Distribution field but that’s irrelevant. In undergrad we were taught about all of the transforms but not really in any practical sense. I know they can be used to analyze signals and stuff but I’m Not entirely sure what use they serve. So if you take derivations or integrations of various formulas you come up with practical formulas for other quantifiable values that exist in the universe. EX: position vs time, velocity vs time, acceleration va time, jerk, snap crackle pop etc. So is there any benefit to running common formulas you learn in school, F=ma, V=ir, KE=0.5mv^2 etc through Laplace transforms? Like does it land you at a significant general formula for another quantifiable value ?

I understand up to setting (1/X)(d^2X/dx^2) = C1 and (1/Y)(d^2Y/dy^2) = C2. How do we know it's then k^2X and -k^2Y? Then, it becomes X(x) = Ae^kx + Be^-kx and Y(y) = Csinky +Dcosky?

How do we come to each conclusion and how would we generalize it so we can use it for 3 dimensions as well. Up until now it's just been a memorized thing, but I can't grasp this unless I understand how everything is derived.

I have a webwork question that is driving me crazy. All of the parts are correct but I don't know what I should enter for the blank part.

https://preview.redd.it/e6l7gw4h78861.png?width=1277&format=png&auto=webp&s=32e624378a94a639bce481be97b5d425d711278e

Edit: I talked to my lecturer. It seems like the question is asking the shift that is made by e^(7t) part. So the answer is s-7.

I want to solve the Laplace equation in plane polar coordinates (r,θ) for r > 1, subject to the boundary conditions:

u(1, θ) = f(θ)

and

lim_{r→∞) u(r,θ) = 0.

The general solution bounded at infinity can be shown, via a separation of variables method, to be

u(r,θ) = a₀ + ∑₁^(∞) r^(-n) ( a_n cos(nθ) + b_n sin(nθ) ).

So we can match coefficients of the Fourier expansion of f(θ) to get a_n and b_n. However, we see that u doesn't decay to infinity unless a₀ = 0. This is essentially a restriction on the boundary data, f(θ); f cannot have a constant in its Fourier expansion for the solution to decay at infinity.

My question is this: is there a way to get around this problem, perhaps by expanding the constant a₀ as a Fourier series of a square wave over a larger domain? If not, can someone provide an argument as to why the constant at r = 1 must be the same as the constant at infinity? Is this a case of too much boundary data? How much data needs to be specified for the solution of a PDE on the plane to be guaranteed to exist and be unique?

Thanks in advance!

Good day everyone, i've been recently working in E&M problems where i found potentials through the solution of laplace equation (given some boundaries conditions). My problem is, all of this exercises involve only two zones where i find the potential (for example inside and outside of a charged sphere or inside and outside of a magnetized cylinder) and recently i have encountered problems where there's 3 zones involved (for example a thick spherical shell magnetized between its two radius) and i have struggled a lot to apply boundary conditions with these kind of problems (i know that it should work the same, but honestly i don't get it). If someone could give an explanation (or even an example) of how these kind of problems work i would really apreciate it. Thanks for reading.

If the stream function of a flow satisfies the Laplace equation, what does this imply about the flow?

If the velocity potential of a flow does not satisfy the Laplace equation, what does this imply about the flow?

Hey all, I’m currently self studying PDE’s in order to place out of it at my university. My university uses Haberman’s Applied Partial Differential Equations text, but the only version of it online is the second edition, whereas my university uses the 5th edition. One of the units covered in the 5th edition that is not present in the edition I have is the idea of solving PDE’s in spherical coordinates, as well as legendre polynomials and their formulation. Does anyone have any recommendations on where to learn more about this topic? I can’t find much on YouTube or the internet (most websites I’ve found are more geared towards using legendre polynomials to solve physical problems in electrostatics or quantum mechanics) and so I was wondering if any of you had any recommendations (preferably free) on where to learn the theory behind this topic. Thanks in advance!

https://imgur.com/jhtb11y

I've attached my work and what 2 correct solutions could be. Thanks for any help you can provide.

I’ve been reading Griffiths recently, and the method of solving Lapland’s equation via separation of variables caught my eye.

The series solution we get for the potential, can we use that in most other ways we can use a Taylor series for example? Assuming the terms converge fast enough, could we use the first term and calculate the gradient to get an approximate expression for the electric field?

I have a final coming up covering chapter 3 of Griffiths Electrodynamics. Does anyone know of a good lecture/youtube video that covers it well?

So I've been trying to help a friend with a laplace transform equation, and for the life of me I can not begin to understand how to solve the last segment in this problem. There's two functions for the transform, so the multiplication rule applies right? for lack of being able to send special characters this is the equation, and the work/context as well
https://imgur.com/a/XarEdMh
Please help!
Thanks in advance!

https://preview.redd.it/16p7fm71utd61.png?width=869&format=png&auto=webp&s=0d3b4ba77677c2286c7d510d82b046621afa7749