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. S

    Non homogeneous linear system

    Homework Statement Verify that the given vector is the general solution of the corresponding homogeneous system and then solve the nonhomogenous system. Assume that t>0. tx' = |2 -1|x + |1- t^2| |3 -2| |2t | General solution: x = c1| 1|t + c2| 1|t^-1 | 1|...
  2. M

    Understanding Homogeneous Differential Equations

    i am having trouble trying to classify whether or not a differential equation is homogeneous. i know that if it is, f(tx, ty) = tnf(x, y) but i don't really know what this means for example, why is f(x, y) = exy not homogeneous but f(x, y) = ex/y is homogeneous?? does that mean a...
  3. M

    Homogeneous differential equation

    i have the equation x2y' = (y2 - xy) which i changed to (y2 - xy)dx + x2dy = 0 i then tried to solve using the substitution y = xv dy = xdv + vdx so .. (x2v2) dx + x2(xdv + vdx) = 0 (v2 - v) dx + xdv + vdx = 0 v2dx + xdv = 0 (1/x)dx + (1/v2) dv = 0 ln|x| + C - (1/v) = 0 y(x) =...
  4. I

    Linear Algebra - homogeneous equation

    Homework Statement The problem is setting up the equation, it says that the matrix equation will be made up of four equations for the 2 unknowns. I'm supposed to find for which a's and b's the equation is true, using a linear system and gaussian elimination. Homework Equations A2 + aA + bI2 =...
  5. A

    Homogeneous linear differential equations

    Homework Statement The equation 2y'' - y' + y2(1 - y) = 0; where y' = dy/dx and y'' = d^2y/dx^2 represents a special case of an equation used as a model for nerve conduction, and describes the shape of a wave of electrical activity transmitted along a nerve fi bre. Homework...
  6. M

    General solution for homogeneous equation

    i am having trouble finding the general solution for the given homogeneous equation: x2yy' = (2y2 - x2) which i made into x2dy = (2y2 - x2) dx i turned it into the following: (2y2 - x2) dx - x2 dy = 0 then i used substitution of y = xv and got (2(xv)2 - x2 - x2v) dx - x3 dv = 0 then...
  7. R

    Linear, nonlinear, homogeneous, nonhomogeneous PDE's

    1. For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons. (a) ut-uxx+1=0 (b) ut-uxx+xu=0 (c) ut-uxxt+uux=0 (d) utt-uxx+x2=0 (e) iut-uxx+u/x=0 (f) ux(1+u2x)-1/2+uy(1+u2y)-1/2=0 (g) ux+eyuy=0 (h)...
  8. M

    Atom & Molecular Density of 235-U & Water Solution

    Homework Statement homogeneous solution of 235-U and water, contains 10g of 235-U/L of solution, what is the atom density of 235-U and the molecular density of water Homework Equations The Attempt at a Solution first converting 10g/L to 0.01g/cm^3, density of water 1g/cm^3...
  9. P

    How to prove differential property of homogeneous function

    I came across of an equality which I have difficulty to understand. If f_n is a rational algebraic homogeneous function of degree n in the differential operators and if g_n is a regular non-differential homogeneous function of the same degree n, following equality takes place [Hobson: The theory...
  10. srfriggen

    Linear Algebra, solution to homogeneous equation

    Homework Statement The problem has to do with diagonalizing a square matrix, but the part I'm stuck on is this: Bx=0, where B is the matrix with rows [000], [0,-4, 0], and [-3, 0, -4]. After performing rref on the augmented matrix Bl0, I get rows [1,0,4/3,0], [0,1,0,0], [0000]. I am...
  11. R

    Exploring the Subspace of a Homogeneous System of Linear Equations

    Homework Statement Suppose you have points of a specific form, say (x, y, 3x + 2y). Show that this set of points is a solution to a homogeneous system of linear equations, hence a subspace. The Attempt at a Solution I'm wondering how one is able to go about this. Here's my try, but I'm not...
  12. M

    Effects of homogeneous and inhomogeneous magnetic fields on particles

    Hi, To understand the difference between uniform magnetic fields and field gradients would it help to make comparisons between their effects on different particles? The posts on Stern-Gelach shed some light here. For instance, what effect would a homogeneous and an inhomogeneous magnetic field...
  13. D

    Stuck trying to solve a non homogeneous differential equation

    Homework Statement dx/dt=x(1-2y) t(0)=0 x(t(0))=1 dy/dt=-y(1-2x), t(0)=0 y(t(0))=2 inerval of integration= [0,40] Homework Equations The Attempt at a Solution i let x=p(t) so t=(p^-1)(x) dy/dx=(∂y/∂(p^-1))∂(p^-1)/∂x dy/dx=(-y(1-2x))/(x(1-2y)) since this equation is...
  14. M

    How does GR slow a homogeneous universe?

    While the title may suggest this thread belongs in cosmology, my questions are orientated towards a better understanding of general relativity. Basically, there seems to be an assumption that after the Big Bang, the initial expansion of the universe was slowed by gravity. Many sources appear to...
  15. M

    2nd order non homogeneous equation

    Homework Statement y'' + y = -2 Sinx Homework Equations The Attempt at a Solution finding the homogeneous solution, is simple; yh(x) = C1 Cos(x) + C2 Sin(x) for the particular solution, I let y = A Cos(x) + B Sin(x) thus, y' = -A Sin(x) + B Cos(x) y'' = -A Cos(x) - B...
  16. R

    Homogeneous Linear D.E. Solutions: Step-by-Step Guide | Urgent Help

    Homework Statement A linear equation in form: dy/dx + P(x)y = 0 is said to be homogeneous since Q(x)=0. a) show that y=0 is a trivial solution (wasn't even taught what a trivial solution is) b) show that y=y1(x) is a solution and k is a constant, then y=ky1x is also a solution. c)...
  17. N

    Non homogeneous system question

    there is a system Ax=b (x and b are vectors) A is a system of kxn type the system has at leas one solution: 1. if k>n does the system has endless solutions ?? 2. if k=n and Ax=b has a single solution then for any system Ax=c has a solution ?? for 1: i don't know why we need the...
  18. N

    Solution of homogeneous question

    x and y are solutions to an equation system i know that if x and y are solutions to a homogeneous system then for any a b ax+by (linear diversity) is also a solution but does it go the other way around to if x and y are solutions and ax+by also then its a homogeneus system ?
  19. Z

    Solutions of Homogeneous System

    Homework Statement If X0 and X1 are solutions to the homogeneous system of equation AX = 0, show that rX0 + sX1 is also a solution for any scalars r and s. Thanks for help! Homework Equations The Attempt at a Solution
  20. A

    A solution to a homogeneous system

    Homework Statement I know there is a polynomial that is a solution to the equation: 3/2f(x) - x/2f'(x) - f''(x)=x Homework Equations The Attempt at a Solution I tried many polynomials of degrees 1,2 and 3, but they do not work in my equation
  21. L

    2nd order homogeneous equations complex root

    Homework Statement y'' -2y' +5y =0 , y(0)=1, y'(0)=1 you get a complex root conjugate. Homework Equations y=e^(rt) y'=re^(rt) y''=r^2 * e^(rt) The Attempt at a Solution I have in my notes sin(omega*t)e^(sigma *t), cos(omega *t)e^(sigma). I don't think i took down the notes...
  22. P

    Unique solution of 1st order autonomous, homogeneous DE

    Hello, 1st order autonomous, homogeneous differential equation have the general form: \dot{x}(t)=ax(t) It can be shown that the unique solution is always: x(t)=e^{at}x(t_{0}) I don't get this, could anyone help me? Thanks!
  23. E

    Proving homogeneous systems of linear equations have infinite solutions

    Proving a linear system of equations cannot have more than one finite solutions Homework Statement Prove that the number of solutions to a linear system can not be a finite number larger than one. Provide either a general proof or for a system with two equations and two unknowns...
  24. J

    Can Zero Wavefunction Be the Only Solution in Quantum Mechanics?

    Homework Statement Solving the following differential equation with the given boundary conditions: \hbar^2 \frac{d^2}{dx^2}\psi (x) = 2mE\psi (x), \ \ \ \ \ \forall \ \hbar^2,\ m,\ E > 0 \psi(a) = \psi(-a) = 0 Homework Equations The Attempt at a Solution \hbar^2 \frac{d^2}{dx^2}\psi (x)...
  25. Z

    Associated Homogeneous System definition

    Hi, thank you for viewing this thread. I have been googling for its definition for quite a while, but have not found any yet. Just wondering if there is a definition of it, in mathematical notations and in words?
  26. B

    Solving 2nd Order Homogeneous Equations with Non-Constant Coefficients

    Hi Guys, I know how to find the solution to a 2nd order homogeneous with constant coefficients but how do you solve one with a non constant ie x^2y''+2xy' ... etc = 0 Is there a general solution formula for these types of problems? My book seems to jump from 2nd order...
  27. P

    Set of vector which are solution to 2 homogeneous systems

    hi guys, I have two homogeneous systems S1 and S2. The solution for NS(S1) = {[-2,1,0], [-1,0,1]}, NS(S2) = {[-2,1,0], [-3,0,1]}. I know that in a system if u and v are vectors, the sum of u+v is also a solution in the homogeneous system. i.e. S1=span{[-2,1,0], [-1,0,1]} then [-2,1,0] +...
  28. FeDeX_LaTeX

    Solving Difference Equations with Homogeneous and Inhomogeneous Parts?

    Hello; This is not a homework question, but something I was wondering about solving difference equations. For example, how would I solve the following difference equation; F_{n} = 2F_{n - 2p + 5} + 6p - 17; n, p \in \mathbb N Since it has homogeneous and inhomogeneous parts together (and two...
  29. Y

    Homogeneous Fredholm equation of the second kind

    Hi, during the analysis of a problem in my phd thesis I have resulted in the following equation. \varphi(x)= \int_a^b K(x,t)\varphi(t)dt which is clearly a homogeneous Fredholm equation of the second kind The problem is that I can't find in any text any way of solving it. Solutions are...
  30. R

    Homogeneous Laplace's Equation

    Homework Statement uxx+uyy=0 u(x,0)=u(x,pie)=0 u(0,y)=0 ux(5,y)=3siny-5sin4y Homework Equations The Attempt at a Solution Using separable method I get Y"-kY= 0 and X"+kX=0 For Case 1 and Case 2 where k>0 and k=0 there are no eigenvalues So Case 3 k<0 gives Y=ccos(sqrk...
  31. jaketodd

    Time dilation by homogeneous distribution of massive objects?

    If a low-mass object is surrounded by massive objects homogeneously so that the low-mass object does not experience acceleration, then is there any time dilation due to the gravitation from the massive objects? Thanks, Jake
  32. Y

    Please help is solving the non homogeneous heat problem.

    Homework Statement Find solution of a nonhomogeneous heat problem: \frac{\partial U}{\partial t} = c^2( \frac{\partial^2 U}{\partial r^2} + \frac{1}{r}\frac{\partial U}{\partial r} + \frac{1}{r^2}\frac{\partial^2 U}{\partial \theta^2} + g(r,\theta,t) With boundary condition...
  33. T

    When to ask the homogeneous question

    When determining a particular solution to a differential equation, one of the necessary steps is to ask the "homogeneous question" aka Does any term in yp solve the homogeneous equation for this problem. When it does, I know that it is necessary to multiple by t. My question is, do I multiply...
  34. T

    Basis for the homogeneous system

    Homework Statement Find a basis for the solution space of the homogeneous systems of linear equations AX=0 Homework Equations Let A=1 2 3 4 5 6 6 6 5 4 3 3 1 2 3 4 5 6 and X= x y z...
  35. R

    What is exact difference btwn ISOTROPIC and HOMOGENEOUS materials

    what is exact difference btwn ISOTROPIC and HOMOGENEOUS materials (kindly don't tell definition of those things)
  36. A

    Homogeneous differential equation

    Homework Statement (1-xcotx)y''-xy'+y=0 y1(x)=x is a solution find the second solution, y2(x), y1 and y2 are linear independent Homework Equations N/A The Attempt at a Solution i only know how to find it by auxiliary equation by substitute y=erx and also i can't use substitution y=xr...
  37. D

    Please check my work: verify that the equation is homogeneous

    Homework Statement verify the equation is homogeneous and solve.. thanks (x^2 - 2y^2)dx + (xy)dy = 0 (x^2 - 2y^2) = -(xy)dy/dx (x^2 - 2y^2)/-(xy) = dy/dx -(x/y) - 2(y/x) = dy/dx v = (y/x) .. y = vx and dy/dx = v + xv' -(1/v) - 2v = v + xv' x(dv/dx) + (1/v) +...
  38. J

    A Homogeneous Linear System w/ Constant Coefficients

    I'll make this post short. The problem just asks me to something in the form x'=Ax (A is a 2x2 of constants) and then describe the behavior of the solution as t approaches infinity. My solution is x=C1e-2t(2/3 1)T + C2e-t(1 1)T. Since both vectors are multiplied by 1/et, my solution...
  39. J

    2nd order non homogeneous diff. eqs. 2nd posting for clarification

    I recently attempted to solve the following: y” + (K/m)y = (Kl^{0}+mg)/m y(0) = l_{0} y(t_{e}) = (K l_{0}+mg)/K The Attempt at a Solution y(t) = -(mg/K)cos{\sqrt{K/m} t} + (mg/K){cos{\sqrt{K/m} t_{e}}/sin{\sqrt{K/m} t_{e}}}*sin{\sqrt{K/m} t} + (K l_{0}+mg)/K which...
  40. M

    Thermodynamics: showing that U (energy) is a homogeneous function of S,V and N

    We know that S (entropy) is additive and satisfies the relation: \lambdaS(U,V,N)=S(\lambdaU,\lambdaV,\lambdaN) (S is a homogeneous function of U, V and N) I need to show that U is a homogeneous function of S, V and N that is, to show that \lambdaU(S,V,N)=U(\lambdaS,\lambdaV,\lambdaN) I...
  41. P

    Differential Equation (Homogeneous / scale-invariant

    Homework Statement Test the following equation to show that they are scale invariant. Find their general solutions (It is not necessary to do the anti-derivative.) (x+y^2)dy+ydx=0 I believe what my tutorial wants me to do is to check for homogeneity. I'm not sure though. This is not a...
  42. H

    Stable Nuclei present during Homogeneous Nucleation

    Homework Statement Assume for the solidification of nickel that nucleation is homogeneous, and the number of stable nuclei is 10^6 nuclei per cubic meter. Calculate the critical radius and the number of stable nuclei that exist at the following degrees of supercooling: 200 K and 300 K...
  43. L

    First Order Homogeneous Equations

    Homework Statement \frac{dy}{dx} = \frac{3xy}{3x^2+7y^2}, y(1)=1 Express it in the form F(x,y)=0 The Attempt at a Solution I'm not sure where I'm going wrong. I let v=y/x, v+x\frac{dv}{dx} = \frac{3x^2v}{3x^2+7x^2v^2}=\frac{x^2(3v)}{x^2(3+7v^2)}= \frac{3v}{3+7v^2} \Rightarrow...
  44. R

    Homogeneous Differential Equation

    (Moderator's note: thread moved from "Differential Equations") Hello :) I'm trying to solve an homogeneous equation... but it seems that I'm wrong in some step... or something, because I can't complete this problem, look here is what i got: x\frac{dy}{dx} = ye^\frac{x}{y} - x x dy =...
  45. N

    Green's function for homogeneous PDE

    Hi there, could anyone help me on this particularly frustrating problem I am having... I have a linear parabolic homogeneous PDE in two variables with a boundary condition that is a piecewise function. I can solve the pde (with a homogeneous BC) however trying to impose the actual BC makes...
  46. S

    Differential Equations- homogeneous (I think)

    Homework Statement Find the general solution of the equation x*y*(dy/dx)=(x^2) + 3(y^2)Homework Equations The Attempt at a Solution So I start by realizing this is (likely) a homogeneous differential equation, and then rewrite it in the form required: dy/dx = (x/y) + 3(y/x) then, using the...
  47. K

    General solution to a second order homogeneous differential equation

    Homework Statement Find if it is true that the general solution to : y'' - y' = 0, where y(x), can be written as : y(x) = c1 cosh(x) + c2 sinh(x), where c1 and c2 are real arbitrary constants. Homework Equations differential equation solving The Attempt at a Solution I just...
  48. J

    Differential equation help solving homogeneous equations

    Homework Statement Find a general solution if possible, otherwise find a relation that defines the solutions implicity. xy'-y=x tan(y/x) Homework Equations The Attempt at a Solution y'-(y/x) = tan(y/x) v = y/x ; y = xv ; y' = xv' + v xv' + v = tan(v) + v xv' =...
  49. K

    Homogeneous and inhomogenous relaxation time

    Consider two-level system, the relaxation time (T1) and the coherence relaxation time (T2). I wonder what's the relation between T1, T2 in homogeneous and inhomogeneous case? Here is my thoughts. For inhomogeneous case, all atoms are behave independently, the 'random' phase relation will add...
  50. M

    Solving a Homogeneous Linear ODE: (x^2-1)y'' + 4xy' + 2y = 6x

    Homework Statement Solve: (x^2-1)y'' + 4xy' + 2y = 6x, given that y_1=\frac{1}{x-1} and y_2=\frac{1}{x+1}. Homework Equations The Attempt at a Solution Since both solutions are given, the solution to the homogenous system is: y_h=C_1\frac{1}{x-1} + C_2\frac{1}{x+1} And the...
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