Partial Definition and 1000 Threads

Homework Statement Find the solution of each of the following partial differential equation \frac{\partial^{2}u}{\partial x^{2}} = 0 Homework Equations assume the product form? u(x,y) = f(x)g(y) ? (not 100% sure) The Attempt at a Solution Hello, I'm only new to PDEs and I was...

Why is it assumed that the method of separation of variables works when the boundary conditions of some boundary valued problem are homogeneous? What is the reasoning behind it?

When would you use trig substitution vs. partial fractions? I know partial fractions is when you have a polynomial over a polynomial, but some of the problems in the trig substitution section in my book had polynomial over polynomial and used trig substitution?

I've come across using partial derivative notation for taking the partial derivative of a function f with respect to a vector x. I've never seen this before. It is also being referred to as a gradient. However, I have only seen gradients where all variables in the space are featured in the...

In a problem that requires converting from cartesian to polar coordinates, I need to take \frac{dr}{dx}. I tried doing it two different ways but getting two completely different answers.. Method 1: r=\sqrt{x^2+y^2} \frac{dr}{dx}=\frac{1}{2}\frac{1}{\sqrt{x^2+y^2}}2x \;\; =...

Homework Statement Suppose z=ψ(2x-3y), Show that the second partial derivative of z with respect to x, is equal to the second partial derivative with respect to y multiplied by a scalar k. Homework Equations The Attempt at a Solution I thought this was too simple to be correct...

Suppose we have a function V(x,y)=x^2 + axy + y^2 how do we write \frac{dV}{dt} For instance if V(x,y)=x^2 + y^2, then \frac{dV}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} So, is the solution \frac{dV}{dt} = 2x \frac{dx}{dt} + ay\frac{dx}{dt} + ax\frac{dy}{dt} + 2y \frac{dy}{dt}

Homework Statement \int \frac{-2x + 4}{(x-1)^{(2)}(x^{(2)}+1)}Homework Equations The Attempt at a Solution I've done the problem a couple times but the answers keep coming out differently so I'm assuming I am messing up the setup. This is what I have for the first part of the setup: -2x +...

Hi guys, Question is: Find the slopes of the curves of intersection of surface z = f(x,y) with the planes perpendicular to the x-axis and y-axis respectively at the given point. z = 2x2y ...at (1,1). fx(x,y) = 4xy ∴ Slope = 4 fy(x,y) = 2x2 ∴ Slope = 2 Is this wrong? Answer...

Hello, Wolfram is giving me the required answer however, the steps it uses I find very confusing. Can anyone share some light on how wolfram achieved the correct answer. As I am new to this site, I won't be using any code. I am in the process of writing it up on Latex. Here is the link...

Homework Statement Find the partial fractions expansion in the following form, G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2} Homework Equations The Attempt at a Solution I expanded things out and found the following, 1 = A(s^{2} + 4)...

Homework Statement (50 - x - y)(x+y) - x - ((y^2)/2)) simplified: 50X - X^2 - XY - 50Y - XY - Y^2 - X - ((Y^2)/2)Homework Equations 50X - X^2 - XY - 50Y - XY - Y^2 - X - ((Y^2)/2)) The Attempt at a Solution for x: 49 - 2x - 2y for y: 50 - 2x - 3y The book has the same, except -1x and...

Hello, PF! As I was reading my P-Chem textbook, I noticed most thermodynamic equations involve partial derivatives, like these ones: C_V = {\left( \frac {\partial E}{\partial T} \right )}_V {\left( \frac {\partial H}{\partial T} \right )}_P = {\left( \frac {\partial E}{\partial T} \right )}_P +...

Homework Statement Evaluate ∫((secx)^2)/[((tanx)^2)+(3tanx)+2] Homework Equations Partial fraction decomposition The Attempt at a Solution So here's what I did: But this is incorrect. It says the correct answer is -2lnabs(\frac{1}{2tanx+3}+\sqrt{4(tanx+3/2)^{2}-1}), which was...

Hello, I am working on the derivation that proves that the partial molar volume of an ideal gas is equal to the molar volume of an ideal gas. I am following up to the point in the textbook where they set (∂n/∂ni)nj = 1 where ni is the number of of moles of species i, and nj is the...

Homework Statement Homework Equations After looking through this on Wiki, I'm a little confused as to how these partial fractions are multiplied out. Is there a rule or something for this? With simpler partials I can do it but this one is something else! The Attempt at a Solution

Homework Statement f = \frac{1}{z(z-1)(z-2)} Homework Equations Partial fraction The Attempt at a Solution R1 = 0 < z < 1 R2 = 1 < z < 2 R3 = z > 2 f = \frac{1}{z(z-1)(z-2)} = \frac{1}{z} * (\frac{A}{z-1} + \frac{B}{z-2}) Where A = -1 , B = 1. f = \frac{1}{z} *...

Hello all, I have this function here: \[f(x,y)=\left\{\begin{matrix} z &(x,y)\neq (0,0) \\ 0 & (x,y)=(0,0) \end{matrix}\right.\] where \[z=\frac{x^{3}+xy^{2}}{2x^{2}+y^{2}}\] And I need to find it's first partial derivative by x and y at the point (0,0). I am not sure I know how to approach...

Hey! :o I have to find the Normal form of the 2.order partial differential equation. I am not sure if my solution is correct.. The differential equation is: $ u_{xx}-4u_{xy}+4_{yy}-6u_x+12u_y-9u=0$ $a=1, b=-2, c=4$ $b^2-ac=4-4=0 \Rightarrow $ parabolic $\frac{dy}{dx}=\frac{1}{a}(b \pm...

You can give me a good examples where ##\frac{\partial}{\partial x}## is different to ##\frac{d}{dx}## ?

Hi! :smile: I have the following integral \int^{∞}_{∞} \frac{\delta^{n}}{\delta a}f(a,b,c)da there is any way to rewrite it in terms of: \int^{∞}_{∞} f(a,b,c)da I want to evaluate it for the case of n=1,2 and 3. Thanks you so much.

Okay so the partial fraction decomposition theorem is that if f(z) is a rational function, f(z)=sum of the principal parts of a laurent expansion of f(z) about each root. I'm working through an example in my book, I am fine to follow it. (method 1 below) But instinctively , I would have...

Homework Statement use partial fraction decomposition to re-write 1/(s2(s2+4) The Attempt at a Solution I thought it would break down into (A/s) + (B/s2) + ((cx+d)/(s2+4) but it doesn't.

I am not quite sure how \frac{\partial}{\partial u}\left(\frac{\partial z}{\partial u}\right) =\frac{\partial}{\partial u}\left( u \frac{\partial z}{\partial x}+v\frac{\partial z}{\partial y} \right) comes to \frac{\partial z}{\partial x} + u\frac{\partial}{\partial u}\left(\frac{\partial...

Like we have the total differential of a function: I was thinking, why not take the "total integral" of a function too? Thus I did some algebraic juggling and, how I haven't aptitude for be a Ph.D. in math, I bring my ideia for the experients from here evaluate... Anyway, the ideia is the...

Homework Statement If possible, please check my work for any large errors. y = 10kl - √k - √l k = (t/5) + 5 l = 5e^t/10 Evaluate at t = 0 using chain rule. Homework Equations y = 10kl - √k - √l k = (t/5) + 5 l = 5e^t/10 The Attempt at a Solution = ∂y/∂k * dk/dt + ∂y/∂l * dl/dt = (10l -...

When I'm evaluating a problem like $$ \int \frac{2x^2 + 8x + 9}{(x^2 + 2x + 5)(x + 2)} = \frac{Ax + B}{x^2 + 2x + 5} + \frac{C}{x+2}$$ I understand how to get the C part, that's simple. But what is a Good trick to know that I need to have $$Ax + B$$ over the $$x^2 + 2x + 5$$ denominator? Is...

Homework Statement Using the formal limit definition of the derivative, derive expressions for the Fourier Transforms with respect to x of the partial derivatives \frac{\partial u}{\partial t} and \frac {\partial u}{\partial x} . Homework Equations The Fourier Transform of a function...

\int (x+1/x2-3x-5)dx I can't put the limits on the integral sign, 5 is the top limit and 3 is the bottom limit. I can solve using partial fractions ok but I have never solved with limits before. Where do the limits come in, do I need them at the start or can I factorise as usual and use...

Good afternoon guys! I have some doubts about partial derivatives. The other day, my analytic geometry professor told us that slopes do not exist in three-dimensional space. If that's the case, then what does a partial derivative represent? Given that the derivative of a function with respect to...

Hello! :) Having the transformations: $$\xi=\xi(x,y), \eta=\eta(x,y)$$ I want to find the following partial derivatives: $$\frac{\partial}{\partial{x}}= \frac{\partial}{ \partial{\xi}} \frac{\partial{\xi}}{\partial{x}}+\frac{\partial}{\partial{\eta}}...

I am trying to figure out the deflection in a fully restrained beam. A diagram of the beam can be found here on this website. http://civilengineer.webinfolist.com/fb/fbcalcu.php I have been able to find the reactionary forces as well as the moments at each end of the beam for any distributed...

Problem: I did some of the problem on MatLab but I'm having a difficult time evaluating the derivatives at (0,0). Also, MatLab gave me the same answer for fxy and fyx, which according to the problem isn't correct. Any ideas? I used MatLab and computed: fx(x,y)=(2*x^2*y)/(x^2 + y^2) + (y*(x^2 -...

Homework Statement Homework Equations The Attempt at a Solution Umm can somebody explain to me what just happened. None of that makes any sense to me what so ever.

Homework Statement 1/ (x+8)(x^2+16) Find the integral Homework Equations I keep getting this question wrong. Can someone check my steps? The Attempt at a Solution I set it up as A/(x+8) + (Bx+C)/(x^2+16) So I did, A(x^2+16)+ (Bx+C)(x+8) and I did that and got A+b=0...

Homework Statement (2x^3-2x+1)/(x^2/3x) Find the integral. 2. The attempt at a solution So I've been on this problem for like an hour now and I don't know what I'm doing wrong. So I used long division and got 2x+ (4x+1)/(x^2-3x) ∫2x + ∫(4x+1)/(x^2-3x) = x^2 +...

Homework Statement I(alpha) = ∫1/((x+alpha^2)(x+1)) dx between the limits of 0 and infinity Evaluate the integral above depending on the parameter alpha using partial fractions. The Attempt at a Solution 1/((x+alpha^2)(x+1)) = A/(x+alpha^2) + B/(x+1) 1 = A(x+1) + B(x+alpha^2)...

I am trying to separate out \[ \frac{s}{(s+1)^3} \] for an inverse Laplace transform. How does one setup up partial fractions for a cubic? I know for a square I would do \[ \frac{A}{s+1} + \frac{Bs+C}{(s+1)^2} \] I tried doing \[ \frac{A+Bs}{(s+1)^2} + \frac{Cs^2+Ds+E}{(s+1)^3} \] which led to...

Homework Statement ∫(5x2+20x+6)/(x3+2x2+x Homework Equations The Attempt at a Solution (5x2+20x+6)/(x3+(x(x2+2x+1) (5x2+20x+6)=(A/x)+(B/(x+1))+(C/(x+1)) (5x2+20x+6)=x2(A+B+C)+x(2A+B+C)+A 5=A+B+C 20=2A+B+C 6=A It's not coming out quite right. Did I maybe factor the...

Homework Statement ∫(2x3-4x-8)/(x2-x)(x2+4) dx Homework Equations The Attempt at a Solution ∫(2x3-4x-8)/x(x-1)(x2+4) dx Next I left off the integral sign so I could do the partial fractions: 2x3-4x-8=(A/x)+(B/(x-1))+((Cx+D)/(x2+4))...

Homework Statement Use integration by parts to evaluate the integral ∫(7-6x) / (x2-4x+13)The Attempt at a Solution This is a question from my notes so I already have the solution but I'm not sure what's going on at this one specific step. ∫(7-6x) / (x2-4x+13) = -∫(6x-7) / (x2-4x+13) = -∫(...

Homework Statement If ##xs^2 + yt^2 = 1## (1) and ##x^2s + y^2t = xy - 4,## (2) find ##\frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \frac{\partial y}{\partial t}## at ##(x,y,s,t) = (1,-3,2,-1)##. Homework Equations Pretty much those just listed...

Use integration to find a solution involving one or more arbirary functions \frac{\partial u}{\partial y}=\frac{x}{\sqrt{1+y^2}} for a function u(x,y,z) u(x,y,z)=x\int \frac{dy}{\sqrt{1+y^2}} let y=\sinh v u(x,y,z)=x\int \frac{\cosh v\: dv}{\sqrt{1+\sinh ^2v}} u(x,y,z)=x\sinh ^{-1}y+f(x,z)...

Ok I'm stuck I have $$\int \frac{x^2 - 5x + 16}{(2x + 1)(x - 2)^2} \, dx$$ and I got to this part: $$x^2 - 5x + 16 = A(x - 2)^2 + B(x - 2)(x + \frac{1}{2}) + c(x + \frac{1}{2})$$So do i need to distribute all of these and factor out or is there a simpler way? I found a solution where they are...

Hello, first post here. Here is a hypothetical partial pressure of gas question Imagine you have a two component system: Component 1 is water Component 2 is an imaginary substance that is immiscible with water, but has same boiling temp / pressure You put the two components in a...

Here is the question: I have posted a link there to this thread so the OP can view my work.

Hello MHB members and friends!(Callme) An economy student asked me, if I could explain the following partial differentiation: \[\frac{\partial}{\partial C(i)}\int_{i\in[0;1]}[C(i)]^\frac{\eta - 1}{\eta}di =\int_{j\in[0;1]}[C(j)]^\frac{\eta - 1}{\eta}dj\frac{\eta -...

Quick question... I know that if the numerator is greater than the denominator I need to divide out by long division BUT If the numerator is equal to the denominator (the exponent is what I'm talking about to be specific) then, do I need to do anything? Because I'm stuck on this problem $$\int...

Homework Statement (t4+9)/(t4+9t2) Homework Equations The Attempt at a Solution I'm not completely sure if I'm using the correct method to solve this. Since the degrees of the numerator and denominator are the same, wouldn't you divide the denominator into the numerator? Here is...

Homework Statement A function f(x,t) depends on position x and time t independent variables. And if \dot{f} represents \frac{df(x,t)}{dt} and \dot{x} represents \frac{dx}{dt}, then find the value of \frac{\partial\dot{f}}{\partial\dot{x}}. Homework Equations The Attempt at...