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. K

    Solving a Second Order Homogeneous Differential Equation

    I just finished a final in my differential equations class. One of the problems had me solve a second order homogeneous differential equation using series. I boiled it down to this recursion relation: a_{n+2}=\frac{(n+3)a_{n}}{2(n+2)(n+1)} I found that the even coefficients work out...
  2. D

    Homogeneous second order linear ODE

    Hey; not much of a homework question, but something i was wanting to find out. Im still a first year undergraduate and just started on differential equations. We have just finished going over homogeneous 2nd order ODE's of the form: ay'' + by' + cy = 0 My texbook briefly outlines the...
  3. S

    Second Order Linear Homogeneous differential equations

    hey I am having a little trouble with this topic. Here are the questions I was set. a) Find the general solution of d^2y/dt^2 - 2(dy/dt) + y = 0. Verify your answer. b) Solve the initial value problem y'' + 4y' + 5y = 0; y(0)= -3, y'(0) = 0 c) Find a DE that has the given functions as...
  4. P

    Solutions to Homogeneous 2nd-order Linear D.E.s

    ,so in class we're covering second-order D.E.s... How exactly was it determined/derived that all solutions to homogeneous 2nd-order linear differential equations with constant coefficients (ay'' + by' + cy =0, where a,b,c are real constants) are of the form y=ert. Not asking for anyone to...
  5. A

    2nd Order Linear Diff. Eqn (homogeneous)

    Homework Statement Show that id y = x(t) is a solution of the diff. eqn. y'' + p(t)y' + q(t)y = g(t), where g(t) is not always zero, then y = c*x(t), where c is any constant other than 1, is not a solution. Homework Equations Can someone help me get started? Also, since g(t) is not...
  6. E

    Non Homogeneous Recurrence Relation

    1. solve the following recurrence relation for an 2. (n+2)an+1= 2(n+1)an+2^{n}, n>=0, a0=1 I shifted the index, multiplied through by the 2^{n} term and then subtracted the resulting equation from the original equation to get rid of the 2^{n} term... 3. I have gotten to this point...
  7. S

    Intro to PDE: Related homogeneous boundary condtions

    (partial derivatives didn't carry over well, so I just used a d) Homework Statement Give an example (as simple as possible) of a reference temperature distribution r = r(x, t) satisfying the following boundary conditions DN: r(0, t) = A(t), (dr(L,t) / dx) = B(t); NN: (dr(0,t) / dx) =...
  8. J

    Discontinuities of coefficient in linear homogeneous system

    Homework Statement let x^{(1)} = \left( \begin{array}{ccc}1\\1\\1\end{array} \right) , x^{(2)} = \left( \begin{array}{ccc}1\\t\\t^2\end{array} \right) , x^{(3)} = \left( \begin{array}{ccc}1\\t\\t^3\end{array} \right) a) Find the Wronskian W(x^{(1)}, x^{(2)}, x^{(3)}) b) You are...
  9. G

    Hamiltonian for homogeneous system

    I have to show that the hamiltonian for a homogeneous system can be simplified in scaled coordinates. The first two terms I can convert to scaled coordinates <T>+<V> whereas I have some trouble for the last term -½* \int d³r d³r' \frac{n²}{|r-r'|} where n is the density. The scaled...
  10. G

    Gen.Chem: homogeneous vs heterogeneous

    Homework Statement Classify each of the following as a pure substance, solution, or heterogeneous mixture: a) blood b) dry ice c) krypton gas d) a rusty nail e) table salt f) glass of lemonade Justify your answers. Homework Equations No equations obviously...The Attempt at a Solution Here's...
  11. D

    Electromagnetism Homogeneous Wave Equation

    I've managed to derive from Maxwell's equations the homogeneous electromagnetic wave equation with respect to the magnetic field. (The one that goes Del Squared of H minus (The second order partial derivative of B multiplied by the recipricol of C squared all equal to zero) Hopefully that...
  12. M

    Solve Homogeneous Ladder Problem - Max Height Reachable

    Homework Statement A homogeneous ladder having a length of 6m and a mass of 35kg is positioned. The lower base of the ladder has an angle of 60 degrees with the ground and the coefficient of friction between former and the latter is 0.3. Define the maximum height a human can reach, which...
  13. T

    Proof on homogeneous equations

    I need some help here... I've got the following assignment to do Prove that if M>N then any system of N homogeneous equations in M unknowns has many solutions. I am a bit stuck with this one. I thought about creating a MxN Matrix and to display the determinant with 1's. and then say...
  14. S

    Question about homogeneous dif. equations

    I've been confused for a little while as my teacher in dif. eq.'s taught us about homogeneous equations, where you can call an eq. homogeneous if all terms are of the same degree. But then, when seeing linear dif. equations we were taught that an homogeneous eq. is one with certain form and...
  15. S

    II order homogeneous differential equation solution

    I am trying to solve the diff. equation - a\frac{d^{2}x}{dr^{2}} + (br + c/r)\frac{dx}{dr} + dx = 0 I got it while solving a variant of damped harmonic oscillation. Any hints (Frobenius method won't work)
  16. I

    Second Order DE: Nonlinear Homogeneous

    I am sure most of you are familiar with the equation: m(x)''+c(x)'+k(x) = 0. Then, we create an auxillary equation that looks like this: mr^2+cr+k = 0. And, then we find the roots of this auxillary equation, calling them r1 and r2. And, if the roots are r1,r2>0 we consider the system to be...
  17. Y

    Imperfectly conducting sphere spinning in a homogeneous B-field

    Homework Statement Link to assignment The fact that it is imperfectly conducting is supplied so that the charges in the sphere will move with the same angular velocity. The B-field induced by the moving charges will be disregarded. Homework Equations...
  18. K

    Reduction of order (2nd order linear ODE homogeneous ODE)

    Reduce to the 2nd order and solve the eq. (1-x^2)y'' - 2xy' + 2y = 0 (we know that y(1) = 0) I don't know what to do here, except that I'm supposed to find y(2). I know the formulas y(2) = y(1) Int(U) dx where U = (e^(-Int(p(x)dx)) / (y(1))^2 Can I use this?
  19. M

    Homogeneous nonlinear function

    Could you give me an example of a function that satisfies scalar multiplication but not addition? more specifically, F: R^2 -> R such that F(av)=a F(v) but F(v1 + v2) != F(v1) + F(v2) The best thing I could come up with is F(x,y)= |x| . This obviously does not satisfy additivity, but...
  20. N

    Homogeneous and Isotropic Stress-Energy Tensor in Cosmology

    This is a very basic question but what is the difference between isotropic and homogeneous? I mean, I can imagine a universe that would be isotropic but not homogeneous although this seems to select a preferred frame (am I worng on this?) But I don't understand how it would be possible to have a...
  21. W

    Second order homogeneous Differential EQ with complex coefficients.

    Homework Statement This is the result of a problem from my Quantum class, but I figure it would be best to ask in here as my question is purely a question of how to solve a certain differential equation. the equation is of the form 0=Y''-i*a*Y' + b*Y where Y is a function of t So the...
  22. B

    Help with linear homogeneous recurrence relations

    Homework Statement Here's my problem - Give the order of linear homogeneous recurrence relations with constant coefficients for: An = 2na(n-1) The Attempt at a Solution I have no idea on how to start this problem - Any help would be greatly appreciated.
  23. S

    Nonlinear Homogeneous Differential Equation Solution

    Homework Statement I'm told that this is homogenous (x^2-xy)y'+y^2 = 0 2. The attempt at a solution This is going to be very painful for me to type out... (x^2-xy)\frac{dy}{dx} = -y^2 \frac{dy}{dx} = \frac{-y^2}{(x^2-xy)} \frac{dy}{dx} = \frac{-y^2}{(x^2-xy)}...
  24. S

    Understanding Homogeneous Differential Equation Simplification

    This is a basic simplification, but I'm going to post this here because it becomes homogeneous, and I know v = \frac{y}{x} but I don't see this simplification, I don't understand how it gets from this... \frac{dy}{dx} = \frac{y-x}{y+x} To THIS: = \frac{v-1}{v+1} (I'm just only showing...
  25. D

    Why Can an Equation Be Homogeneous but Incorrect?

    im posed with the question why an equation may be homogeneous with respect to its units but still be incorrect? i can't think of way to explain this can anyone help me out? Thanx
  26. P

    Difference between spatially homogeneous and isotropic?

    What are their differences? Spatially homogeneous is when there is uniform composition of space Spatially isotropic is when you look anywhere, they look the same Is it the case that one is visit anywhere, it is the same and the other is look anywhere they look the same? They seem...
  27. T

    Solving Homogeneous Problem: Initial Value & Substitution

    Hi all, I have been given this question: Find the initial value problem of the homogeneous equation: (x^2 - y^2) y' = xy \ , \ y(1) = 1 Now I know, from my lessons, I have to get it in the form of: \frac{dy}{dx} = f(\frac{y}{x}) I have managed to get close but nothing is...
  28. D

    Solve homogeneous first order PDE's

    u_x+u_y+u=e^{x+2y}, y(x,0)=0 I have no idea how to do this. We were only taught how to solve homogeneous first order PDE's.
  29. D

    ODE particular solution of the homogeneous equation

    y''(t)+A^2y(t)=f(t), t>0, y(0)=B, y'(0)=C, A, B, C\in\mathbb{R} e^{iAt} is a particular solution of the homogeneous equation. I can multiply it by some arbitrary function and find another solution of the homogeneous case, but when I try with the f(t) on the RHS, I can't do it. Anyone help?
  30. D

    Showing Integration Factor of Homogeneous Equation of Degree n

    Hey... So the question is as stated: Show that \frac{1} {M_x + N_y} , where M_x+N_y is not identically zero, is an integrating factor of the homogeneous equation M(x, y)dx+N(x, y)dy=0 of degree n. So I am not too sure where to go with this. I suppose what it's saying is, that I'm...
  31. K

    Higher Order Homogeneous ODE (IVP)

    Higher Order Homogeneous ODE (IVP) [Solved] I am having problems with this IVP: y'''' + y' = 0 y(0) = 5 y'(0) = 2 y''(0) = 4 What I have done so far is: \lambda^3 + \lambda = 0 \lambda(\lambda^2 + 1) = 0 So one roots is \lambda = 0 (though.. can there be a root that...
  32. Z

    Homogeneous Differential Equation

    Solve the initial value problem y''' - 5y'' + 100y' - 500y = 0; y(0) = 0, y'(0) = 10, y''(0) = 250 given that y_1(x) = e^5x is one particular solution of the differential equation. r^3 - 5r^2 + 100r - 500 = 0 r = +/- 10i or 5 complementary solution y_c = e^5x(c1cos10x + c2sin10x) general...
  33. A

    Differentiating Homogeneous Functions: A & B

    Well actuallly 2 thms. They have to do with homogeneous functions. f(tx1,...,txn) = t^k * f(x1,...,xn). Now how do you show A) d/dx1 f(tx1,...,txn) = t^k-1 * d/dx1 f(x1,...,xn) and B) kt^(k-1)*f(x1,...,xn) = x1*d/dx1 f(tx1,...,xn) + xn*d/dxn f(x1,...,xn) A) In the book They say that...
  34. M

    Question on homogeneous linear systems

    I have a homework question that I don't really understand what they are asking. The book I am using is terrible so I was hoping someone could shed some light. Question: Give a geometric explanation of why a homogeneous linear system consisting of two equations in three unknowns must...
  35. A

    Solving a Homogeneous Linear Equation for x>0

    Can someone give suggestions for this question? Find a basis of solutions for the following second-order homogeneous linear equation for positive x: x^2y``-xy`+y=0
  36. T

    Homogeneous system of linear equation:

    I am having trouble finding the solution to the homogeneous system of linear equations: 2x-2y+z=0 -2x+y+z=0
  37. F

    Solving a Homogeneous ODE: Step-by-Step Guide

    Hi, I'm having trouble with this ODE: y' + 4xy - y^2 = 4x^2 - 7 => y' = 4x^2 - 4xy + y^2 - 7 = (2x - y)^2 - 7 => x(dv/dx) + v = (2x-vx)^2 - 7 = x^2(2-v)^2 - 7 I assume this ODE is of the homogeneous type, so I've substituted v=y/x and gotten thus far...
  38. S

    Why must every homogeneous system of planes have at least 1 intersection point

    Well, I'm doing homework (again). I was introduced to homogeneous systems of planes and then asked why there must be at least 1 intersection point. The book gives very little (one sentence) on homogeneous systems so I tried to search around online. My guess is that since all of the...
  39. G

    Homogeneous and particular solution

    hi, I have this ODE and I need to obtain the general and the particular solution, this is the ODE Vz''+1/r*Vz'= k where k is a constant thanks
  40. L

    Understanding S^2 as the Quotient of SO(3) and SO(2)

    I've seen S^2 written as the quotient SO(3)/SO(2). Can someone run me through how to show this, or point to somewhere that does, as I've only seen it stated?
  41. N

    Finding a homogeneous solution

    Q. Determine a homogeneous linear differential equation with constant coefficients having having the following solution: y = C1sin3x + C2cos3x My idea is to differntiate both sides with respect to x and come up with an equation in dy/dx what else? can be done... Is my idea correct.
  42. A

    Second-Order Homogeneous Linear Equation

    I have this intial value problem: y''-4y'-5y=0, y(1)=0, y'(1)=2. My AUX equation is r^2-4r-5=0. I factor and get r=5, r=-1 and my equation becomes y(x)=C1e^(5x)+C2e^(-1x) (C1 and C2 are constants). I took the derivative of y(x) and then tried to use my initial value's to solve for C1 and C2. I...
  43. C

    How do I prove the homogeneity of a function of degree n?

    Hi " A function f is called homogeneous of degree n if it satisfies the equation f(tx,ty,tz)=t^n*f(x,y,z) for all t, where n is a positive integer and f has continuous second order partial derivatives". I don't have equation editor so let curly d=D I need help to show that...
  44. N

    Homogeneous Differential Equations

    Hi, I need some help in finding whether this differential equation is homogeneous or not. 3 (d^2 y / dx^2) + x (dy/dx)^2 = y^2 I know that for example, x^2 dx + xy dy = 0 is homogeneous. But how can I deal with the equation that has (d^2 y / dx^2) and (dy/dx)^2 ? Thanks
  45. G

    Show that the equation is homogeneous

    Show that the equation is homogeneous with respect to units: I = nAQv I can't prove it, please help
  46. K

    How Does Substitution Work in Solving Homogeneous Differential Equations?

    The book only has one example of this and it's really confusing me. (x^2+y^2)dx+(x^2-xy)dy=0 I can see that it's homogeneous of degree 2 They then let y = ux From there they state that dy = udx+xdu (I'm not sure where this is coming from, but can just accept it on faith if I have to)...
  47. cepheid

    General Solutions of 2nd Order Linear Homogeneous Ordinary D.E.s

    A theorem in my textbook is confusing me: For the functions p(t) \ \ \text{and} \ \ q(t) continuous on an open inteval I defined by \alpha < t < \beta : We have differential equation L[y] = 0 where L = (\frac{d^2}{dt^2} + p\frac{d}{dt} + q) The theorem attempts to prove...
  48. A

    What is the solution for the homogeneous equation (x^2 + y^2)dx + (2xy)dy = 0?

    (x^2 + y^2)dx + (2xy)dy = 0 I get y = sqrt((kx^5 + x^2)/3) Where k = c2 cubed, and c2 = ln(c) so k = 3ln(c) But, the answer the teacher gave is (x^2)(y^3) - x - ln(y) = c I can't come up with anything remotely close. I know this isn't in a pretty LaTeX form, but I am new and haven't...
  49. cepheid

    Is Homogeneous Always Separable in First Order Differential Equations?

    Hi, Please bear with me, I've only had the first sort of "pseudo-lecture" in ordinary d.e.'s this past week, and I was doing some reading ahead. It occurred to me that if linear first-order differential equations are those that can be written in the general form: \frac{dy}{dx} + P(x)y =...
  50. L

    Homogeneous linear system question

    Hi, i have a question. Hope you guys can help~ Ques: Give a geometric explanation of why a homogeneous linear system consisting of 2 equations in 3 unknowns must have inifinitely many solutions. What are the possible numbers of solutions for a nonhomogeneous 2 x 3 linear system? Give a...
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