What is Homogeneous: Definition and 402 Discussions

Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, weight, height, distribution, texture, language, income, disease, temperature, radioactivity, architectural design, etc.); one that is heterogeneous is distinctly nonuniform in one of these qualities.

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  1. M

    I "Rationale" for Homogeneous vs. Nonhomogeneous Differential Equations?

    Hi; I am missing something. I can follow the technicality of a homogenous linear equation has all coefficients of zero and the "contra" for non homogenous equations. I just can't figure out the relevance of the consequences of outcome. If I am not being clear maybe I can be guided as to how...
  2. K

    B Pressure in a viscous liquid versus non-viscous fluids

    Hc verma, concepts of Physics, vol 1 pg 258 "We define pressure of fluid at the point A as : ##P= F/\Delta S## For a homogeneous and non-viscous fluid, this quantity does not depend on orientation of ##\Delta S## and hence we talk of pressure at a point". Why did the author stress that the...
  3. B

    I Questions about algebraic curves and homogeneous polynomial equations

    It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...
  4. V

    A Cosmological Density Perturbation vs Homogeneity: Questions Answered

    When arriving at the standard model of cosmology, i.e. the exapnding universe, we assume based on experirmental data that the cosmos is homogenous on large enough scales. But when we go back in time, when the galaxies are beginning to form, we note that because of the growth of density...
  5. JD_PM

    A Non-isothermal conditions in liquid-gas nitrogen homogeneous mixture

    I am studying the cavitation theory proposed by...
  6. O

    Show that ODE is homogeneous, but I don't think it is

    Ignoring the second part of the question for now, since I think it will be more clear once I understand how this equation is homogeneous. According to my textbook and online resources a first-order ODE is homogeneous when it can be written like so: $$M(x,y) dx + N(x,y) dy = 0$$ and ##M(x,y)##...
  7. LCSphysicist

    When do we need to consider the homogeneous solution?

    Homework Statement:: All below Relevant Equations:: , Generally, when for example we need to solve ##\nabla u = 0##, we separate variables and find equations like that ##X''/X = -Y''/Y = k^2##. So we just solve it, sum the solutions and make it satisfy the boundary/initial conditions. But...
  8. K

    A Dissipation function is homogeneous in ##\dot{q}## second degree proof

    We have Rayleigh's dissipation function, defined as ## \mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right) ## Also we have transformation equations to generalized coordinates as ##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...
  9. karush

    MHB -2.2.35 Show that dy/dx=(x+3y)/(x-y) is homogeneous. and....

    $\dfrac{dy}{dx}=\dfrac{x+3y}{x-y}$ ok well following the book example: divide numerator and denominator by x $\dfrac{dy}{dx}=\dfrac{1+3\dfrac{y}{x}}{1-\dfrac{y}{x}}$ apparently, thus this is homogeneous but not sure why? next solve the DE:unsure:
  10. chwala

    Solve the homogeneous ODE: dy/dx = (x^2 + y^2)/xy

    this is pretty easy for me to solve, no doubt on that. My question is on the constant. Alternatively, is it correct to have, ##ln x= \frac {v^2}{2}##+ C, then work it from there... secondly, we are 'making" ##c= ln k##, is it for convenience purposes?, supposing i left the constant as it is...
  11. P

    A De-Sitter Spacetime: Is it Homogeneous & Isotropic?

    The question is in the title. I believe the answer is yes.
  12. mitochan

    I Coincidence of FLWR & CBR Homogeneity: Earth @ 0.0013c?

    The Earth is moving with respect to the CBR at a speed of 390 kilometers per second, I read in the article https://www.scientificamerican.com/article/how-fast-is-the-earth-mov/. Does FLWR metric coordinate space coincides with integrated local FRs where CBR is homogeneous, and the Earth is...
  13. AndreasC

    Non-interacting gas in homogeneous gravitational field

    It even gives a hint, it says "consider two horizontal surfaces z1 and z2 and think about what thermodynamic equilibrium means for particles traveling from one surface to the other". This really trips me up because I am not sure what to do with this. Obviously in equilibrium the number of...
  14. Wizard

    A Parametric Lagrangian is a Homogeneous Form in Parametric Velocities?

    In the book "The Variational Principles of Mechanics" by Cornelius Lanczos, the following statement is made about a lagrangian ##L_1## where time is given as an dependent parameter, and a new parameter ##\tau## is introduced as the independent variable, see (610.3) and (610.4) pg. 186,187 Dover...
  15. A

    Help with solution group of a Homogeneous system

    Summary:: need help with solution group of Homogeneous system Is the solution group of the system A^3X = 0 , Is equal to the solution group of the system AX = 0 If this is true you will prove it, if not give a counterexample. thank you.
  16. A

    Equilibrium temperature in a homogeneous section (stationary regime)

    I believe it is just an arithmetic medium, but I am not sure. Could someone explain to me?
  17. karush

    MHB 311.1.5.5 homogeneous systems in parametric vector form.

    Write the solution set of the given homogeneous systems in parametric vector form. $\begin{array}{rrrr} -2x_1& +2x_2& +4x_3& =0\\ -4x_1& -4x_2& -8x_3& =0\\ &-3x_2& -3x_3& =0 \end{array}\implies \left[\begin{array}{rrrr} x_1\\x_2\\x_3 \end{array}\right] =\left[\begin{array}{rrrr}...
  18. O

    A Inhomogeneous wave equation: RHS orthogonal to homogeneous solutions

    Hi, I've been reading Brillouin's 'Wave Propagation in Periodic Media'. About the following equation $$\nabla^2u_1+\frac{\omega^2_0}{V_0}u_1 = R(r)$$ Brillouin states that "it is well known that such an equation possesses a finite solution only if the right-hand term is orthogonal to all...
  19. P

    Confirming Green's function for homogeneous Helmholtz equation (3D)

    Plugging in the supposed ##G## into the delta function equation ##\nabla^2 G = -\frac{1}{4 \pi} \frac{1}{r^2} \frac{\partial}{\partial r} \left(\frac{r^2 \left(ikr e^{ikr} - e^{ikr} \right)}{r^2} \right)## ##= -\frac{1}{4 \pi} \frac{1}{r^2} \left[ike^{ikr} - rk^2 e^{ikr} - ike^{ikr} \right]##...
  20. karush

    MHB -b.2.2.33 - Homogeneous first order ODEs, direction fields and integral curves

    $\dfrac{dy}{dx}=\dfrac{4y-3x}{2x-y}$ OK I assume u subst so we can separate $$\dfrac{dy}{dx}= \dfrac{y/x-3}{2-y/x} $$
  21. karush

    MHB -b.2.2.32 First order homogeneous ODE

    \[ \dfrac{dy}{dx} =\dfrac{x^2+3y^2}{2xy} =\dfrac{x^2}{2xy}+\dfrac{3y^2}{2xy} =\dfrac{x}{2y}+\dfrac{3y}{2x}\] ok not sure if this is the best first steip,,,, if so then do a $u=\dfrac{x}{y}$ ?
  22. karush

    MHB -2.2.31 First order homogeneous ODE

    I OK going to do #31 if others new OPs I went over the examples but? well we can't 6seem to start by a simple separation I think direction fields can be derived with desmos
  23. Diracobama2181

    Potential and E field for a non homogeneous charge Density

    Based on the conditions, I found that $$V(x)=\frac{a^2}{\pi^2} ρ_0sin(πx/a)$$ would be a solution to Laplace's equation for $$|x|\leq a$$ and $$V(x)=cx+d$$, where c and d are constants. From the boundary conditions, $$\frac{dV(a)}{dx}=\frac{a}{\pi} ρ_0cos(πa/a)=ac$$, $$c=\frac{a\rho}{\pi}$$ and...
  24. Moara

    Electron moving inside a region of homogeneous electric field

    a) since the eletric field is perpendicular to the inicial velocity, the x component is constant, hence Vf.cos45=Vo. This gives Vf=0,6√2.C b) Ei=γi.Eo , γi=5/4 , Ef=γf.Eo , γf=5/(2√7) Finally, Ei+e.E.d=Ef. Apparently this is incorrect, why??
  25. H

    I Energy resolution of a homogeneous calorimeter

    Hi I want to look at the energy resolution of a homogeneous calorimeter, in the literature I found that is given by σ/E = a ⊕ b/√E ⊕ c/E and I found that the ⊕ means quadratic sum, what does quadratic sum mean? Thanks Aaron
  26. kunalvanjare

    How can I achieve homogeneous in-line mixing of a Coolant with water?

    Hello guys, This is for a new product I am designing, which involves topping-up or refilling the liquid into a CNC Machine or Part Washer. I have to mix about a Litre of the chemical (alkaline.. pH in the range of 8-12) with about 50 Litres of water. The water will be fed under gravity through...
  27. cianfa72

    I Gaussian elimination for homogeneous linear systems

    Hi, I ask for a clarification about the following: consider for instance a 10 x 12 homogeneous linear system and perform Gauss elimination for the first 8 unknowns. Suppose you end up with 5 equations in the remaining 12-8 = 4 unknowns (because in the process of the first 8 unknowns elimination...
  28. dRic2

    I Homogeneous equation and orthogonality

    Hi, I'm going to cite a book that I'am reading Can anyone provide some simple references where I can find at least an intuition regarding what is stated by the author. Thanks, Ric
  29. karush

    MHB -m30 - 2nd order linear homogeneous ODE solve using Wronskian

    2000 Convert the differential equation $$\displaystyle y^{\prime\prime} + 5y^\prime + 6y =0$$ ok I presume this means to find a general solution so $$\lambda^2+5\lambda+6=(\lambda+3)(\lambda+2)=0$$ then the roots are $$-3,-2$$ thus solutions $$e^{-3x},e^{-2x}$$ ok I think the Wronskain...
  30. PainterGuy

    I Centroid of homogeneous lamina region R and the factor of "1/2"

    Hi, In one of the standard calculus textbooks, source #1, the formula for y-coordinate of center of gravity for a homogeneous lamina is given as follows. In another book of formulas, source #2, the formula is given without the factor "1/2" as is shown below. Personally, I believe that source...
  31. karush

    MHB -a.3.2.96 Convert a 2nd order homogeneous ODE into a system of first order ODEs

    given the differential equation $\quad y''+5y'+6y=0$ (a)convert into a system of first order (homogeneous) differential equation (b)solve the system. ok just look at an example the first step would be $\quad u=y'$ then $\quad u'+5u+6=0$ so far perhaps?
  32. Lone Wolf

    Finding the center of mass of a homogeneous object

    The object is: My attempt at a solution: I divided the object into 3 different rectangles and found the coordinates for the center of mass of each one, considering the origin at point "O". Then I found the mass of each rectangle, assuming the object has an area density of σ. m1 = 15σ; m2= 6σ...
  33. Phys pilot

    MATLAB Plot a non homogeneous DPE -- Sum two plots in Matlab?

    Hello, I want to plot this PDE which is non homogeneous: ut=kuxx+cut=kuxx+c u(x,0)=c0(1−cosπx)u(x,0)=c0(1−cosπx) u(0,t)=0u(1,t)=2c0u(0,t)=0u(1,t)=2c0 I have a code that can solve this problem and plot it with those boundary and initial conditions but not with the non homogeneous term...
  34. Y

    Problem concerning a mass with charge in a homogeneous electric field

    I know how the answer is C, since E=F/q and F=ma=mg. However, I am a bit confused as to why my other method doesn't work. I thought that since the droplets are falling at a constant velocity, there is not net force, so according to E=F/q the electric field must be zero then? This seems like a...
  35. BesselEquation

    Homogeneous Differential Equation

    Homework Statement Solve the following differential equation: y' = y / [ x + √(y^2 - xy)] 2. The attempt at a solution Using the standard method for solving homogeneous equations, setting u = y/x, I arrive at the following: ± dx/x = [1±√(u^2-u) ]/ [u√(u^2-u)] which in turn, I get the...
  36. stephchia

    Finding the linear mapping between homogeneous coordinates

    Homework Statement If I have an affine camera with a projection relationship governed by: \begin{equation} \begin{bmatrix} x & y \end{bmatrix}^T = A \begin{bmatrix} X & Y & Z \end{bmatrix}^T + b \end{equation} where A is a 2x3 matrix and b is a 2x1 vector. How can I form a matrix...
  37. V

    I How can an expanding Universe look homogeneous?

    Observation shows that the Universe is homogeneous (and isotropic) at the large scale, while one expects to see inhomogeneity (increasing density at greater distances) on the past light cone due to expansion. This seems inconsistent. Am I misunderstanding something here?
  38. J

    MHB Solving homogeneous equation

    Hi, I have solved this ODE till half way and got stucked on the integration of some weird expression. Need help for this. Thank you!
  39. J

    MHB Solving Homogeneous Equation

    Hi, I have attached part of my steps for solving the homogeneous equation. The equation is proven to be homogeneous. However after using substitution of y=zx and its' derivative, I was not able to separate the variables conveniently as shown. Please advise. Thank you!
  40. Robin04

    First-order homogeneous recurrence relation with variable coefficient

    Homework Statement I need to find the explicit formula for the following recursive sequence: ##v_n=\frac{2}{1+q^n}v_{n-1}## where ##0<q<1## is a constant Homework Equations I found the following method to solve it...
  41. T

    Water / steam homogeneous mixture

    We all here (I presume that members here have better understanding of physical processes than average person) know that it isn't a fact that water began to boil at 100°C but much before that. When being heated in an open pot, as the temperature rises, water began to boil and more and more water...
  42. R

    Self-inductance of a toroid with a rectangular cross section

    I have found answers on how to calculate the self-inductance of toroid of rectangular cross section, however my question says that "The winding are seen as a thin homogeneous currentlayer around the core" (excuse the translation). What does that mean for N? Does it mean N=1?
  43. G

    I Wire loop in a static homogeneous field question

    I already did a similar question here but got very little response so I will try to reformulate my question into a better one.So, the basic idea of the original question was whether a Faraday disc aka homopolar generator be made such as to have no sliding contacts and the load being attached to...
  44. BookWei

    I Spacetime is homogeneous and isotropic

    I read the Special Theory of Relativity in Jackson's textbook, Classical Electrodynamics 3rd edition. Consider the wave front reaches a point ##(x,y,z)## in the frame ##K## at a time t given by the equation, $$c^{2}t^{2}-(x^{2}+y^{2}+z^{2})=0 --- (1)$$ Similarly, in the frame ##K^{'}## the wave...
  45. T

    MHB Homogeneous, underdetermined equation system

    Hi! Just started with linear algebra Could someone help me with this problem? 2x_1 + x_2 - x_3 + 3x_4 - 3x_5 = 0\\ 3x_1 + 2x_2 + x_3 + 2x_4 + 2x_5 = 0\\ -4x_1 + 3x_2 + 2x_3 + x_4 - 4x_5 = 0 (Sorry, I don't know how to do these big brackets for equation systems in Latex.) So it's a...
  46. A

    A How to simplify the solution of the following linear homogeneous ODE?

    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...
  47. R

    Polarization charge density of homogeneous dielectric

    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...
  48. Arman777

    Homogeneous Diff. Eqn Finding Solution

    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...
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