What is Homogeneous: Definition and 403 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. B

    Solving DE with Homogeneous Sub: v=y/x

    Homework Statement Consider the DE (x + y)y′ = x − y. (a) Solve the DE using the homogeneous substitution v = y/x. An implicit solution is acceptable. (b) We can rearrange the DE into the differential form (y − x) dx + (x + y) dy = 0. Is this equation exact? If so, find an implicit...
  2. M

    Understanding Homogeneous Systems - Expert Answers and Guidance

    Hi all I have question in homogeneous . could please check my answer ?
  3. D

    ODE Proof (2nd order linear homogeneous equations)

    Homework Statement Suppose u, v are two linearly independent solutions to the differential equation u''+p(x)u'+q(x)v=0. If x0,x1 are consecutive zeros of u, then v has a zero on the open interval (x0,x1) Homework Equations The Attempt at a Solution I'm trying to use the...
  4. V

    Significance of the Huge-LQG for homogeneous universe

    I've poked back through the past few weeks of threads, and I've only seen two posts (here and here) with very little discussion commenting about the recent Huge-LQG discovery (http://mnras.oxfordjournals.org/content/early/2013/01/07/mnras.sts497.full). I was hoping some cosmologists could...
  5. B

    Differential Equations, Homogeneous equations

    Homework Statement Use the method for Homogeneous Equations to slove (xy + y^2) dx - x^2 dy = 0 Homework Equations The Attempt at a Solution I attempted to get dx/dy on one side and substitute but could not get farther than this dx/dy = x^2/(xy + y^2)
  6. Fernando Revilla

    MHB Non homogeneous linear differential equation

    I quote an unsolved question posted in MHF (November 7th, 2012) by user NumberMunhcer.
  7. sergiokapone

    Homogeneous gravitational field and the geodesic deviation

    In General Relativity (GR), we have the _geodesic deviation equation_ (GDE) $$\tag{1}\frac{D^2\xi^{\alpha}}{d\tau^2}=R^{\alpha}_{\beta\gamma\delta}\frac{dx^{\beta}}{d\tau}\xi^{\gamma}\frac{dx^{\delta}}{d\tau}, $$ see e.g...
  8. K

    Homogeneous ODE system, how to solve using WOLFRAM

    Hi. If I have a homogeneous ODE with constant coefficient system in the form of 2x2 matrix: X'=A X, A is a 2x2 matrix. How do I solve this using wolfram or matlab?
  9. D

    MHB Solving Homogeneous System $R(r)=A_m\mathcal{J}_m(kr)+B_m\mathcal{Y}_m(kr)$

    $R(r) = A_m\mathcal{J}_m(kr) + B_m\mathcal{Y}_m(kr)$ $$ \begin{pmatrix} \mathcal{J}_m(ka) & \mathcal{Y}_m(ka)\\ \mathcal{J}_m(kb) & \mathcal{Y}_m(kb) \end{pmatrix} \begin{pmatrix} A_{m}\\ B_{m} \end{pmatrix} = \begin{pmatrix} 0\\ 0 \end{pmatrix} $$ In order for our system to have a non-trivial...
  10. DryRun

    Homogeneous Linear ODE with complex roots

    Homework Statement I'm trying to understand the simplification of the general solution for homogeneous linear ODE with complex roots. Homework Equations In my notes, i have the homogeneous solution given as: y_h (t)= C_1 e^{(-1+i)t}+C_2e^{(-1-i)t} And the simplified solution is given as: y_h...
  11. M

    MHB Differential equations: Homogeneous Linear Equations

    Differential equations Homogeneous Linear Equations case 1 +example - YouTube There are more on my channel and will be posting more daily.
  12. K

    Based loop groups as homogeneous spaces

    Homework Statement Let G be a compact connected Lie group define the loop group and the based loop group as LG = \{ \gamma \in C^\infty(S^1,G) \}, \Omega G = \{ \gamma \in LG : \gamma(e_{S^1}) = e_G \} (choose whatever identification of the circle S^1 you like ). Show that \Omega G is a...
  13. L

    Showing this Euler's equation with a homogeneous function via the chain rule

    Homework Statement Ok I have this general homogeneous function, which is a C^1 function: f(tx,ty)=t^k f(x,y) And then I have to show that this function satisfies this Euler equation: x\frac{\partial f}{\partial x}(x,y)+y\frac{\partial f}{\partial y}(x,y)=k\cdot f(x,y) Homework...
  14. chisigma

    MHB Another second order non homogeneous ODE....

    Four days ago on mathhelpforum.com the user ssh [I don’t know if he the same as in MHB…] has proposed the following second order complete linear ODE… $\displaystyle y^{\ ''} – \frac{2+x}{x}\ y^{\ ’}\ + \frac{2+x}{x^{2}}\ y = x\ e^{x}$ (1) … and till now no satisfactory solution has been...
  15. S

    MHB Solving Second order non - homogeneous Differential Equation

    How to solve \( (x+1) y'' - (2x+5) y' + 2y = (x+1) e^x\) can we assume \(y_1 = (Ax+B) e^x \), then \(y_2= vy_1​\) Is this right? then solve for A and B Finally \( y = c_1 y_1 + c_2 y_2\)
  16. iVenky

    Is the Cauchy Equation Still Homogeneous if X is a Non-zero Function of x?

    The Cauchy homogeneous linear differential equation is given by x^{n}\frac{d^{n}y}{dx^{n}} +k_{1} x^{n-1}\frac{d^{n-1}y}{dx^{n-1}} +...+k_{n}y=X where X is a function of x and k_{1},k_{2}...,k_{n} are constants. I thought for this equation to be homogeneous the right side should be 0. (i.e.)...
  17. iVenky

    Is this an homogeneous equation?

    Actually I can't find if a differential equation is homogeneous or not I thought homogeneous is given by dy/dx= f(x,y)/ g(x,y) but it doesn't look like that For eg: dy/dx= (y+x-1)/(y-x+2) is not homogeneous at all though f(x,y)=y+x-1 and g(x,y)=y-x+2 How can you tell...
  18. S

    MHB Solving Second order non - homogeneous Differential Equation

    To Solve y’’ – 2 y’ – 3y = 64 e-x x ---------------(1) Using the method of undetermined coefficients : The roots of the homogeneous equation are 3 and -1, so the complimentary solution is y = c1 e3x + c2 e-x Then the guess for the particular solution of (1) is e-x x (Ax + B)...
  19. A

    Temperature of a homogeneous land surface

    Hello, I have temperature and emissivity data values for different land types (i.e. rock, vegetation etc.) of an area in form of image pixels. Because there are different land types, they show different temperatures. I want to know the temperature of these pixels as if they were of the same...
  20. S

    Isotropic and homogeneous of space

    we say the universe around us is isotropic and homogeneous. it means that all direction and points are the same for some special class of reference. if this is true why we say in large scale universe is isotropic and homogeneous? it seems that the space, itself, to be isotropy and...
  21. L

    2nd order homogeneous diff eq

    Homework Statement y"-2y'+5=0, y(∏/2)=0, y'(∏/2)=2 find general solution of this diff eq Homework Equations The Attempt at a Solution i have followed all of the steps for this, rather easy 2nd order diff eq, but i my solution differs from the books solution. steps...
  22. T

    Can a Homogeneous Equation Still Be Incorrect?

    How is it possible that an equation shown to be homogeneous with respect to its unit may still be incorrect .
  23. C

    Unbounded or infinite would be more appropriate terms to use in this context.

    Homework Statement Find the values of α for which all the solutions of y''-(2α-1)y'+α(α-1)y=0 (a) tend to zero and (b) are ilimited, when t->∞. Homework Equations y''-(2α-1)y'+α(α-1)y=0 => (t)=Ae^{αt}+Be^{(α-1)t} The Attempt at a Solution I found that the general solution to the...
  24. L

    Homogeneous Eqn of Line given 2 homogeneous pointspoints

    I'm reviewing Projective Geometry. This is an exercise in 2D homogeneous points and lines. It is not a homework assignment - I'm way too old for that. Given two points p1 (X1,Y1,W1) and p2 (X2,Y2,W2) find the equation of the line that passes through them (aX+bY+cW=0). (See...
  25. 1

    Help with an (I think) homogeneous DE.

    Homework Statement y' = \frac{2xy + y^{2} + 1}{y(2+3y)} Homework Equations The Attempt at a Solution First I tried making a substitution in the case that it is homogeneous, but it didn't make the equation separable. It's not linear, it's not exact, and not separable. Does it...
  26. S

    Calculating the Inertia Tensor of a Homogeneous Sphere

    Homework Statement Calculate the moments of Inertia I_{1}, I_{2}, I_{3} for a homogenous sphere Homework Equations I_{jk}=\intx^{2}_{l}\delta_{ik}-x_{i}x_{k}dV The Attempt at a Solution For I_{x} i set up the equation using the above equation in cartesian coordinates and then i...
  27. H

    Homogeneous and Heterogeneous assays?

    Could someone explain them?
  28. H

    Infinite solutions to homogeneous system?

    Could someone explain the following theorem to me: Given a homogeneous system of n linear equations in m unknowns if m>n (i.e. there are more unknowns than equations) there will be infinitely many solutions to the system.
  29. B

    2nd order non-linear homogeneous differential equation

    Homework Statement Find a solution (Z2) of: z'' + 2z - 6(tanh(t))2z = 0 that is linearly independent of Z1 = sech2Homework Equations The Attempt at a Solution reduction of order gives you v''(t)(Z1(t))+v'(t)(2 * Z1'(t)) + v(t)(Z1''(t)+p(t)Z1'(t)) = 0 however the third term on the LHS can be...
  30. K

    Proving Homogeneous & Isotropic FRW Universe Energy-Momentum Tensor

    Hi everyone, It's not a real homework problem, but something I am trying to do that I haven't found in the literature. I am still stating the problem as if it was a homework Homework Statement Consider a FRW Universe. That is, ℝ x M, where M is a maximally symmetric 3-manifold, with a RW...
  31. B

    Solve this DE using homogeneous equations

    Homework Statement dy/dx = (6x^(2)+xy+6y^(2))/(x^2) Homework Equations v = y/x y' = v + xv' The Attempt at a Solution y' = tan(6ln(abs(x))-C)/x ===> apparently not correct
  32. R

    An infinite, isotropic, homogeneous, and static arrangement collapses under gravity?

    Can anyone explain this to me? If we have an infinite amount of balls arranged in a kind of cubic matrix, in an infinite and static space...how the heck would that collapse on itself due to gravity? Thanks folks
  33. A

    Homogeneous least squares

    Given a homogeneous linear least squares problem: A^{T}y=0 What is the difference between minimizing y^{T}AA^{T}y (the least square error) and: y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y ? Thanks.
  34. D

    Systems of Homogeneous Linear Differential Equations

    Homework Statement I uploaded the problem statement please see attachment for original problem. The problem number is 4. Homework Equations The Attempt at a Solution For clarity I uploaded what I have done please see the attachment with my work on it. I am not sure if I am doing...
  35. T

    Solving second order linear homogeneous differential equation

    Homework Statement Find the set of functions from (-1,1)→ℝ which are solutions of: (x^{2}-1)\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-4y = 0 Homework Equations The Attempt at a Solution There is a hint which says to use the change of variable: x=cos(θ) doing this I get...
  36. K

    Homogeneous differential equation

    how am i going to determine if a higher order differential equation is homogenous? example, d4y/dx4+d2y/dx2=y d3y/dx3-d2y/dx2=0
  37. T

    Solving second order linear homogeneous differential equation

    Homework Statement Find the set of functions from (-1,1)→ℝ which are solutions of: (x^{2}-1)y''+xy'-4y = 0 Homework Equations The Attempt at a Solution OK, I'm not really sure how to go about solving this equation, I have only previously attempted problems where the functions in...
  38. T

    Solving second order non homogeneous differential equation

    Homework Statement The problem is to solve: y''-2y'+5y = e^{x}(cos^{2}(x)+x^{2}) Homework Equations The Attempt at a Solution I (think I) have solved the associated homogeneous equation: y''-2y'+5y = 0 giving the solution as: y_{h} = e^{x}(C_{1}cos(2x)+C_{2}sin(2x))...
  39. N

    Solving a simple homogeneous linear DE

    I feel that it may be redundant to rewrite the whole problem. I just need to know how the book got from point to point b. The book says that e^{-3x} \frac{dy}{dx} - 3y (e^{-3x}) = 0 is the same as \frac{d}{dx}(e^{-3x}y) = 0 How? I tried dividing and multiplying by some variables to get the...
  40. D

    Offset between non-homogeneous and homogeneous recurrence sequences

    I have a question; help is welcome. Let sn be a linear, non-homogeneous recurrence sequence, and let hn be a corresponding homogeneous sequence (with initial values to be determined). As it turns out, the offset between the two (sn - hn) is given by the steady state value of sn, if the...
  41. A

    Free fall of straight wire in a homogeneous magnetic field

    hello every body. I have a high school problem a straight horizontal wire is falling freely in a homogeneous horizontal magnetic field, perpendicular to the wire and i want to find the inductive voltage. I said E= Blv=Blgt But I can also say E=ΔΦ/Δt=BΔΑ/Δt=Βl1/2gt [t][2]/Δt=1/2Blgt why...
  42. C

    Heat Equation (Non Homogeneous BCs) - Difficult Laplace Transform help ;)

    Heat Equation (Non Homogeneous BCs) - Difficult Laplace Transform... help! ;) Hi I'm trying to model the temperature profile of an inertia friction welding during and after welding. I have the welding outputs and have come up with a net heat flow wrt time during the process. I now want to...
  43. L

    Derive the Integrating Factor for Homogeneous DE

    Homework Statement I have this statement: If M(x,y)dx+N(x,y)dy=0 is a homogeneous DE, then μ(x,y)=\frac{1}{xM+yN} is its integrating factor. The problem is, how do we derive this integrating factor? Homework Equations For homogeneous DE, we have f(kx,ky)=k^n*f(x,y) We also have...
  44. M

    Solving homogeneous system involving decimal eigenvalues

    Homework Statement I need to find the general solution of the system [3 5] [-1 -2] Homework Equations so to get the eigenvalues, det(A - λI) The Attempt at a Solution determinant is (3-λ)(-2-λ) + 5 which would be λ2 - λ - 1 so by the quadratic equation the eigenvalues are...
  45. A

    Doubt about the dimension of a 2nd order homogeneous equation

    My doubt is that is dimension of a 2nd order homogeneous equation of form y''+p(x)y'+q(x)=0 always 2 ? or dimension is 2 only when p(x),q(x) are contionuos on a given interval I..??
  46. S

    MHB Repeated roots, non homogeneous - second order, reduction of order method

    I semi understand the reduction of order method, and i understand the general solution for a 2nd order with repeated roots. however, i can't seem to form up the correct thing to solve this question, and research again proves futile. Any assistance will be appreciated. Use the method of...
  47. S

    MHB Second order homogeneous equations with non constant coefficients

    I was given a question and i am really unsure how to go about solving it. it appears to be solveable using the characteristic equation and whatnot, however i have my coeffecients in terms of the independent variable. so i am confused. the question initially asked to compute the wronskian, and it...
  48. O

    MHB Homogeneous linear ODEs with Constant Coefficients

    do you have a idea about it?can you help me http://img17.imageshack.us/img17/1156/18176658.png
  49. A

    2nd order homogeneous linear diff eq

    Homework Statement y'' + y' - 2y = 0 Homework Equations The Attempt at a Solution I think this is extremely simple. hopefully i am correct. i said the 'auxiliary' equation is r2 + r - 2 = (r+2)(r-1) = 0 the roots are r = 1, -2 so the solution is y=c1ex + c2e-2x correct?
  50. T

    Moment of inertia for non homogeneous density.

    Ok so I am trying to figure how I would find the moment of inertia for a special case. I have a 55 gallon barrel that is almost half way full and I am suppose to spin it roughly 5-10 rpm. I know that to find the momement of interia of a hollow cylinder with thick walls is simply...
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