Homogeneous Definition and 405 Threads

During solution of a PDE I came across following ODE ## \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) ## I have to solve this ODE which I have done using integrating factor using following steps taking integrating factor I=\exp^{\int \frac{1}{D} \alpha^2...

Hi everyone, there's something that I can't comprehend: when a homogeneous is in a conservative and non-uniform in module electric field polarization expression is given by P=ε0χE. Supposing the most general situation there's: divP=ρp where ρp is the polarization charge density in the...

Homework Statement ##(2xy+3y^2)dx-(2xy+x^2)dy=0## Homework EquationsThe Attempt at a Solution It's a homogeneous equation since we can write, ##M(x,y)=(2xy+3y^2)## and ##M(tx,ty)=t^2M(x,y)## and ##N(x,y)=(2xy+x^2)## and ##N(tx,ty)=t^2N(x,y)## since orders of t are same they are homogeneous...

The equation of motion for a charged particle with mass ##m## and charge ##q## in a static magnetic field is: ##\frac{d}{dt}[m{\dot{\vec{r}}}]=q\ \dot{\vec{r}}\times \vec{B}## From this, we can see that ##\frac{d}{dt}[m\dot{\vec{r}}-q \vec{r}\times \vec{B}]=0## and so the following quantity is...

Hello, There are many different wave equations that describe different wave-like phenomena. Being a differential equation, the WE is a pointwise relation and applies to the wavefield at spatial points. The equation is homogeneous when the source term is zero. That means that the solution...

Hello! I was reading an article from Wikipedia(https://en.wikipedia.org/wiki/Lapse_rate) and the formula seems to me not homogenous since g is in m/s2 and cp in J/K=(kg*m2)/(K*s2) so at the end, we'll get K/(kg*m). How they get rid of kg-1 ? Thanks

<Moderator's note: Moved from a technical forum and therefore no template.> Hello, first I'm sorry for my English. I have a problem with my exam task, this answer wasn't done good according to the professor and I have not idea how I can do it in a different way.Write mechanical energy...

Homework Statement I have taken ODE, linear algebra, mechanics, math physics, etc. and we would always go on about how important the homogeneous equation is. To solve for the equation of motion for a harmonic oscillator (for example) we would solve for both the homogeneous and particular...

Consider an infinite homogeneous static universe with a constant mass density $$\rho$$. If we were to calculate the force on a test particle located at a certain point accoring to Newtons law of gravity. It would be logical to conclude from a symmetry argument that the force on the particle...

Homework Statement Let ##n## be some natural number. Solve the following ##n \times n## homogeneous system of equations: $$\sum_{1|i} x_i = 0$$ $$\sum_{2|i} x_i = 0$$ $$\vdots$$ $$\sum_{n|i} x_i = 0$$, where ##a|b## means ##b## is divisible by ##a##. Homework EquationsThe Attempt at a...

Homework Statement I need to solve: x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9 Homework EquationsThe Attempt at a Solution I know that the answer is: y=x^2+2x^3+x^3lnx Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...

Homework Statement [/B] Homework Equations N/A The Attempt at a Solution What I am confused about is where they got the (1/4)mR^2 + (1/12)ml^2 and (1/2)mR^2 from? I am guessing that these came from the integral of y'^2 + z'^2 and x'^2 +y'^2 but I don't understand how this happened exactly...

What is a homogeneous boundary condition? Or, more explicitly, what would make a boundary condition inhomogeneous Many thanks :)

sin(x) * y''(x) + 2cos(x) * y(x) = 0 y(0) = 0 y'(0) = 1 how do I solve it? (I know the solution because I have created the diff. equation starting from a simple function). -- lightarrow

Homework Statement Find the general solution y"+3y'+2y=0 Homework Equations y(t) =c_1e^r_1t + c_2e^r_2t The Attempt at a Solution a=1 b=3 c=2 r^2+3r+2=0 (r+2)(r+1)=0 r_1=-2 r_2=-1 General solution: y(t) =c_1e^(-2t)+c_2e^(-t)I was wondering if the order mattered. The answer in the book is...

It is common is cosmology to study density fluctuations in the early universe. However, it is also common to assume that the background space is homogeneous and isotropic and use the FRW metric. I do not see how density fluctuations can be possible in a homogeneous and isotropic space. Can you...

Homework Statement Show that the homogeneous equation: $$(Ax^2+By^2)dx+(Cxy+Dy^2)dy=0$$ is exact iff 2b=c. Homework Equations None, just definitions. The Attempt at a Solution Let $$M = Ax^2+By^2$$ and $$N = Cxy+Dy^2$$ Taking the partial derivative of M with respect to y and the partial of...

Hello everybody! Can someone please explain me if I may write Maxwell's homogeneous equations with this notation: ∂[λFμν] = 0 thank you. Mick

Homework Statement I'm sorry, I'm not the best in English, but I'll try to translate it. A homogenous rod with length 5.0 m and mass 40 kg, is connected to a pole at 70* at point A. The rod is held in place with a rope that makes 50* with the pole, and is connected to the rod 1.0 m from the...

Homework Statement The electric potential energy v(r) of a charged particle located between two uniformly charged concentric spheres with radii r1 and r2 satisfies the second order differential equation rv′′+2v′=0, r1≤r≤r2 where r is the distance of the charged particle from the common centre...

Given a linear homogeneous 2nd order ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ the general solution is of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{1}(x)$$ where ##c_{1},c_{2}## are arbitrary constants and ##y_{1}(x), y_{2}(x)## are linearly independent basis solutions. How does one prove that...

Homework Statement dy/dx = (x+4y)2 Homework EquationsThe Attempt at a Solution I substitute y=ux, where u is a function of x, and I'm not a ble to solve. My intention was to arrive at a seperable form, but I'm not achieving it.[/B]

I'm trying to solve $\displaystyle x(y-3x)\frac{dy}{dx} = 2y^2-9xy+8x^2$ Let $y = vx$ then $\displaystyle \frac{dy}{dx}= v+x\frac{dv}{dx}$ and I end up with $\displaystyle \log(cx) = \frac{1}{2}\log(y^2/x^2-6y/x+8)$ Is this correct and what am I supposed to do after this?

Please,I am working on the criticality calculation of an homogeneous finite cylindrical reactor core using four-group diffusion equations. I have been able to discretize the multigroup diffusion equations using the finite difference method(FED). But I am stocked on the iterative method to...

From what I've seen so far, the basis of the solution space for all the constant coefficient homo linear DE's have been linear combinations of the exponential function e or of some polynomial multiplied by the exponential function. Is this always true that these DE's always result in solutions...

Mark44 submitted a new PF Insights post Solving Homogeneous Linear ODEs using Annihilators Continue reading the Original PF Insights Post.

Hey! :o I saw in my notes the part that to show the uniqueness we have to prove that $Lx=0$ has only trivial solution. ($L$ is the differential operator) To solve the homogeneous equation $$\sum_{k=0}^m \alpha_k x^{(k)}(z)=0$$ we find the characteristic equation and its eigenvalues...

How would a super-critical heavy water cooled and moderated two fluid aqueous homogeneous reactor with nitrate fuel work? Silicon carbide or alumina can be used as cladding for the internal seed core and blanket walls, with the silicon carbide on the blanket wall cladding stainless steel and...

Hey! :o When we have the non-homogeneous differential equation $$ay''(x)+by'(x)+cy(x)=f(x)$$ and the non-homogeneous term $f(x)$ is of the form $e^{mx}P_n(x)$ we know that the particular solution is $$y_p=x^k(A_0+A_1x+ \dots +A_nx^n)e^{mx}$$ where $k$ is the multiplicity of the eigenvalue...

Homework Statement I need to resolve this with v = y/x dy/dx= (3y2-x2)/(2xy) Homework EquationsThe Attempt at a Solution dy/dx= (3y2-x2)/(2xy) dy/dx= 3y2/2xy -x2/2xy dy/dx = 3y/2x -x/2y dy/dx = 3y/2x - 1/2y/x dy/dx = 3/2 *v - 1/2*v F(v) = 3/2 *v - 1/2*v is that good so far ?

Homework Statement x(dy/dx) - y = sqrt(xy +x2)Homework EquationsThe Attempt at a Solution I got up to this point: u=y/x dy/dx = (sqrt(xy+x2))/x + y/x And then the solution shows this: dy/dx = y/x + (y/x+1)½ Please help, I don't understand how they got to that point.

Homework Statement A homogeneous flexible rope rests on a wedge whose sides make angles α and β with horizontal. The centre of rope lies on C. With what acceleration should the wedge be moved for the rope to stay stationary with respect to wedge? (all surfaces are smooth). Homework Equations...

Homework Statement Regarding the case where the auxillary (characteristic) equation has complex roots, we solve the quadratic in the usual way using i to get the general solution y(x) = e^{\alpha x}\left(C_1 \cos{\beta x} + i C_2 \sin{\beta x}\right) And the textbook shows y(x) = e^{\alpha...

Homework Statement The problem reads: Find a homogeneous linear differential equation with constant coefficients that has the following particular solution: yp = e^(-t) + 2te^(t) + t^(2)e^(t) - sin(3t) Express your equation in differential operator form. (Hint: What annihilators would...

Is Dark matter homogenenius in Universe ? I don't think so but I don't know any idea about it. Thank you

Homework Statement Solve the initial value problem Homework Equations Quadratic Formula The Attempt at a Solution My problem is that I don't understand how to solve the constants now, I understand, 2 equations, 2 unknowns, but when I plug the y(0) = 0 into the YsubH equation...

Hello,I am trying to calculate the velocity in a pipe with length L and Dia D, which is connected to bottom of a pressurized vessel (Vessel dimensions are known, Level of liquid inside the vessel is known). Now i need to figure out the velocity as a function of pressure inside the vessel.We can...

Homework Statement Solve d2θ/dη2 + 2η(dθ/dη) = 0, to obtain θ as a function of η, where θ=(T-T0)/(Ts-T0) EDIT: I should add that this is a multi-part problem, and η is given as η=Cxtm. We had to use that to derive the equation in question above.. So I don't know if this is supposed to be...

http://dwb4.unl.edu/Chem/CHEM869B/CHEM869BInfoFiles/pubCHEM869B-Info005.html In the last question of Quiz 1X Look at lower Ieft hand side. I chose heterogenous mixture but It was given wrong.Why?I chose heterogenous mixture because there are 4 molecules of one substance and only 3 atoms of other...

Hi there! Just say I have large square piece of some homogeneous resistive material like graphite. How would I go about determining the resistance between any two given points? Further, just say I supply a voltage across two arbitrary points, can I determine the voltage difference between any...

Homework Statement Hello, I was just looking for a quick tip: If I have three distinct solutions to a second order linear homogeneous d.e, how would I show that the wronskian of (y1,y2,y3)(x)=0? I know how to show the wronskian is not zero for a linearly independent set, but I'm confused...

I got three equations: l-cm-bn=0 -cl+m-an=0 -bl-am+n=0 In my textbook, its written "eliminating l, m, n we get:" $$ \begin{vmatrix} 1& -c& -b\\ -c& 1& -a\\ -b& -a& 1\\ \end{vmatrix}=0 $$ but if I take l, m, n as variables and since ##l=\frac{\Delta_1}{\Delta}## (Cramer's rule) and...

Homework Statement hello all, Suppose y is a solution of the d.e: y"+p(x)y'+q(x)y= q(x) on the interval (-1,1) with y(0)=1 and y'(0)= 1. What is y? Homework Equations I used the auxiliary equation: m^2+p(x)m+q(x)= q(x) The Attempt at a Solution My question is can I do this? I can cancel...

Can anybody give me a simple and easy example to understand it

Consider the second-order homogeneous linear differential equation $y'' + 4y' + Ky = 0$ Find the general solution if $K = 4$ So here is what I have: $r^2 + 4r + 4 = 0 $ =$(r + 2)(r+2)$ $r=-2$ ? But I thought that you can't do this because you won't be learning anything new if you have two of...

I understand that in a homogeneous electric field, the force on a particle, regardless of its location, is the same. How can this be? Wouldn't a positively charged particle experience a greater force when near the positively charged side? What am I missing?

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \}=\frac{\text{exp}(-\lambda \Delta t)(\lambda \Delta t)^k}{k!}$. And therefore, event count in...

Let ##F : R^n \to R## be a degree-1 positive-homogeneous function. I.e., ##F(\lambda y) = \lambda F(y),## for all real ##\lambda>0## and any nonzero ##y\in R^n##. In this paper, near the middle of p2 at eq(4), the authors introduce $$\ell_a ~=~ \frac{\partial F}{\partial y^a} ~,$$and then they...

Ok, I am really close to this. d-Limonene dissolves Polystyrene (I have tested this) and Liquid Isobutane mixes with d-Limonene (I have also tested this) However, when I mix all three together, the Polystyrene becomes completely separated from the solution. Same thing with all polymer solutions...

why not the 2nd order linear homogeneous ODEs have three Linearly independent solutions or more? I know for the characteristic equation, we can only find 2 answers but.. just wondering if that is the only case to solve the question and if it is, then why it has to be. so my question is,1. 2nd...