What is Homogeneous: Definition and 404 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. 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...
  2. 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...
  3. J

    Conserved quantity for a particle in a homogeneous and static magnetic field

    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...
  4. F

    I Homogeneous Wave Equation and its Solutions

    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...
  5. K

    B Unit Not Homogeneous: Understanding the Formula

    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
  6. B

    Mechanical Energy of a Homogeneous Circular Wheel Rolling at Different Speeds

    <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...
  7. Vitani11

    What does it mean for an equation to be homogeneous?

    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...
  8. H

    I Is the Force on a Particle in a Homogeneous Static Universe Always Zero?

    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...
  9. B

    Solving a Homogeneous System of Equations

    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...
  10. Cocoleia

    Second order non homogeneous ODE, IVP

    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...
  11. I

    Inertia matrix of a homogeneous cylinder

    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...
  12. P

    B What is a homogeneous boundary condition?

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

    I 2° order linear, homogeneous, variable coefficients

    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
  14. M

    ODE homogeneous equations w/constant coefficients

    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...
  15. S

    A Cosmological perturbations in homogeneous and isotropic spac

    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...
  16. MidgetDwarf

    Show that the homogeneous equation (Ax^2+By^2)dx+(Cxy+Dy^2)d

    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...
  17. M

    I Maxwell's Homogeneous Eqns: Notation Explained | Mick

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

    Solving Tension in Rope for Homogeneous Rod

    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...
  19. i_hate_math

    Homogeneous Second Order O.D.E Problem Please help

    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...
  20. F

    I General solution to linear homogeneous 2nd order ODEs

    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...
  21. Aceix

    Solving a homogeneous first-order ordinary differential eqn

    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]
  22. G

    MHB Solving Homogeneous ODE: $\displaystyle x(y-3x)\frac{dy}{dx}=2y^2-9xy+8x^2$

    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?
  23. Babatunde22

    Criticality calculation of an homogeneous finite reactor

    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...
  24. kostoglotov

    Soln space basis for all constant coeff homo linear DE's?

    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...
  25. Mark44

    Insights Solving Homogeneous Linear ODEs using Annihilators - Comments

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

    MHB Exploring Homogeneous Equations and Unique Solutions

    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...
  27. A

    Idea for Aqueous Homogeneous Reactor

    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...
  28. M

    MHB Non homogeneous differential equation - Particular solution

    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...
  29. M

    Homogeneous differential equation - serious help

    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 ?
  30. jdawg

    Can homogeneous substitution solve this differential equation?

    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.
  31. Titan97

    Newton's law problem on homogeneous flexible rope

    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...
  32. kostoglotov

    Question concerning 2nd order homogeneous linear diff eqs

    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...
  33. Alex Ruiz

    Finding a Homogeneous D.E. that has a particular solution

    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...
  34. RyanH42

    Is Dark matter homogeneous in Universe?

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

    2nd Order Homogeneous, Real Roots, Initial Value

    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...
  36. Sivaraman

    Homogeneous equilibrium model-fluid flow

    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...
  37. mudweez0009

    2nd Order, homogeneous Differential Equation

    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...
  38. gracy

    Homogeneous and heterogeneous mixtures

    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...
  39. M

    Electrical Resistance between 2 points in homogeneous plane?

    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...
  40. B

    Wronskian to determine L.D

    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...
  41. AdityaDev

    System of homogeneous equations

    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...
  42. B

    Is My Solution for a Non Homogeneous D.E. with Given Initial Conditions Valid?

    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...
  43. Aafia

    What is isotropic medium and homogeneous medium

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

    MHB Second-order homogeneous linear differential equation

    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...
  45. A

    Force on a particle in a homogeneous electric field

    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?
  46. I

    MHB A math proof within a question about homogeneous Poisson process

    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...
  47. strangerep

    ArXiv:1301.7652 and Euler homogeneous function theorem

    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...
  48. P

    Why Do Polystyrene, d-Limonene, and Isobutane Not Form a Homogeneous Mixture?

    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...
  49. K

    Why does the 2nd order homogeneous linear ODE have 2 general 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...
  50. evinda

    MHB Showing $XF_{X}+YF_{Y}+ZF_{Z}=nF$ with a Homogeneous Polynomial

    Hi! (Smile) Let $F(X,Y,Z) \in \mathbb{C}[X,Y,Z]$ a homogeneous polynomial of degree $n$. Could you give me a hint how we could show the following? (Thinking) $$XF_{X}+YF_{Y}+ZF_{Z}=nF$$
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