Homogeneous Definition and 405 Threads

  1. 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...
  2. D

    First Order Homogeneous Differential Equations

    Homework Statement Find the general solution of the following homogeneous differential equations: (i) \frac{du}{dx} = \frac{4u-2x}{u+x} (ii) \frac{du}{dx} = \frac{xu+u^{2}}{x^{2}} (You may express your solution as a function of u and x together) Homework Equations There are no...
  3. D

    Can someone help me solve this 1st order homogeneous differential equation?

    I was wondering if some-one could give me some advice on how to complete the following equation: \frac{du}{dx} = \frac{4u-2x}{u+x} Whenever I try to complete it I get a function which I don't think should be integratible. Could some light be shed onto this matter?
  4. N

    Non homogeneous differential equation - power series solution

    I am having trouble getting to a solution for this differential equation 2(x^2+2x)y' - y(x+1) = x^2+2x -------- 1 for a series solution, we have to assume y = \sum a_{n}x^n ---------- 2 if we divide equation 1 by x^2 + 2x , we get (x+1)/(x^2+2x) for the y term, which is where my problem...
  5. I

    Can Homogeneous Functions Help Solve Partial Derivative Relationships?

    Hi all, This is a problem I've been working on, off and on, for a few months now. It seems like it should be possible but I just can't figure it out. Suppose you have a first order homogeneous function f(x_1, x_2, x_3). In other words, f has the property that: \lambda f(x_1, x_2, x_3)...
  6. K

    First order ODE, The Homogeneous Method.

    Homework Statement \frac{1}{xy} \frac{dy}{dx} = \frac{1}{(x^2 + 3y^2)} Homework Equations used the substitutions: v = \frac{x}{y} ,and \frac{dy}{dx} = v + x \frac{dv}{dx} The Attempt at a Solution took out a factor of xy on the denominator of the term on the right hand...
  7. D

    Homogeneous Differential equations

    in the following question i am asked to whow that Y1 and Y2 are basic solutions to the homogeneous equations for 4.1) y=ex=y' =y'' xy'' - (x+1)y' + y = x2 xex - (x+1)ex + ex = 0 none of these seem to work, what am i doing wrong?
  8. H

    Proof involving homogeneous functions and chain rule

    Homework Statement A function f is called homogeneous of degree s if it satisfies the equation f(x1, x2, x3,... xn)=t^s*f(x1, x2, x3,... xn) for all t Prove that the \sum from i=1 to n of xi * df/dxi (x1, x2, x3,... xn) = sf(x1, x2, x3,... xn). Homework Equations The Attempt at a Solution...
  9. M

    Constructing a Homeomorphism for Homogeneous Topological Spaces

    Homework Statement For any a \in \left( -1,1 \right) construct a homeomorphism f_a: \left( -1,1 \right) \longrightarrow \left( -1,1 \right) such that f_a\left( a \right) = 0 . Deduce that \left( -1,1 \right) is homogeneous.Homework Equations Definition of a homogeneous topological...
  10. P

    Solving non Homogeneous second order differential equation

    Homework Statement Find the general solution to y'' + y = sec3(x) The Attempt at a Solution Well I can get the characteristic equation: r2 + 1 = 0 r = +-i Then the homogeneous solution is yh = C1excos(x) + C2exsin(x) And I know y = yh + yp but how do I get yp? I've never...
  11. C

    Homogeneous System: Why Invertible A Has No Non-Zero Solutions?

    Hello, In a book I'm reading about linear algebra it's mentioned that in order for the homogeneous system Ax = 0 to have a solution (other than the trivial solution) the coefficient Matrix must be singular. The thing is, I can't remember (the wikipedia page on homogeneous systems didn't turn up...
  12. J

    Electron in Homogeneous Electric Field

    please help me :( Electron inserted in a homogeneous electric field to measure 300 N / C, which directed vertically upwards. The initial velocity of the electron is far 5,00 10^ × 6 m/s and goes to 30 degrees, above the skyline. a) Find the maximum height that reaches the electron above the...
  13. I

    Homogeneous equation (third order)

    Homework Statement Find y as a function of x if y'''−11y''+28y'=0 y(0)=1 y'(0)=7 y''(0)=2 I have one attempt left on this question. Could someone verify my answer for me? Homework Equations The Attempt at a Solution (use t as lamda) t^3-11t^2+28t=0...
  14. J

    2nd order linear differential equation (homogeneous)

    Homework Statement Solve 354y`` −692y` + 235y =0 y(0) = 7 y`(0) = 4 Homework Equations The Attempt at a Solution First I divided the equation by 354 to get y`` - 1.56y` + 0.894y = 0. Then I found the roots of this to be 0.94, repeated twice. For repeated roots the solution looks like y=...
  15. E

    Explaining Time Homogeneous Lagrangian and Hamiltonian Conservation

    if the lagrangian is time homogenous ,the hamiltonian is a constant of the motion . Is this statement correct ?
  16. mnb96

    How to Solve a Homogeneous System with Norm Constraint?

    Hello, how would you solve an homogeneous system of the form A\mathbf{x}=0, with the constrain <\mathbf{x},\mathbf{x}>=1. The matrix A is symmetric, but I don't know if it matters. There should be a method involving eigenvalues, but strangely enough, I can't find it in any book. Thanks!
  17. D

    Understanding Linear Homogeneous Systems: Finding the Correct Answers

    Homework Statement If a linear homogeneous system Ax=0 has a non - trivial solution and A is an n x n matrix, then (choose ALL correct answers) A. A has rank less than n B. Each system Ax=b with the same coefficient matrix A has a solution C. A is row equivalent to I D. If Ax=b has...
  18. T

    Homogeneous Linear DE's - solving IVP's

    Homogeneous Linear DE's -- solving IVP's Homework Statement Solve the given IVP: d^2y/dt^2 - 4 dy/dt -5y = 0; y(1)=0, y'(1)=2 Homework Equations N/A The Attempt at a Solution I've solved and got the general solution y=c1e5t+c2e-t I'm plugging in the following to solve...
  19. M

    First Order Homogeneous Equation

    Homework Statement (4y4-9x2y2-144)dx - (5xy3)dy = 0 Homework Equations substitute y = xv dy = dx v + dv x The Attempt at a Solution after substituting i got (4x4v4-9v2x4-14x4)dx - (5v3x4)dx.v + dv.x = (4v4-9v2-14)dx - 5v3(dx.v + dv.x) = 0 = dx(4v4-9v2-14-5v4)+dv(-5v3x)= 0...
  20. A

    Solve Homogeneous System: Use Determinant to Check Nontrivial Solutions

    how does one use the determinant of the coefficient matrix of a system to determine if the system has nontrivial solutions or not?
  21. K

    Solution space of linear homogeneous PDE forms a vector space?

    Homework Statement Claim: The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space". Proof: Assume Lu=0 and Lv=0 (i.e. have two solutions) (i) By linearity, L(u+v)=Lu+Lv=0 (ii) By linearity, L(au)=a(Lu)=(a)(0)=0 => any linear...
  22. J

    Finding Homogeneous Solutions for Differential Equations: Does Your Guess Work?

    Homework Statement Given a differential equation. ie. y'' +y' + 1 =0 (THIS IS NOT THE PROBLEM THAT I AM SOLVING) Homework Equations No equations The Attempt at a Solution The equation above is not what I'm working with, but an example of a differential equation problem that I was...
  23. J

    Homogeneous Differential Equation with Initial Conditions

    Hi guys, I've been out of the loop with differentials for some time, and was hoping to get some direction with these.. Given that: (x^2)y" + 2x(x-1)y' - 2(x-1)y = 0 A) Explain why the general theory doesn't guarantee a unique soln to the equation satisfying the initial conditions y(0)...
  24. S

    Proving Homogeneous Deformation: From Spheres to Ellipsoids

    Homework Statement Prove that in the homogeneous deformation, particles which after the deformation lie on the surface of a shere of radius b originally lay on the surface of an ellipsoid. Homework Equations homogeneous deformations are motions of the form: xi=ci + AiRXR where ci...
  25. G

    Second Order Homogeneous Eq's, Auxiliary Eq for complex roots - Help

    Hi I'v got a maths exam on Tuesday for my 2nd year of chemial engineering. Been going through a past paper and have been going over 2nd order homogenous DE's Im at the stage of calculating the roots (wether repeated or 2 distinct roots) I take the easy path like so: E.g m2 + 4m + 4 =...
  26. M

    4th order homogeneous linear ODE with constant coefficients

    Can someone explain to to me how to find the general solution of the fourth order ODE y''''-y''=0 Right now I have y(x)=a+b*x+c*e^-x+d*e^x where a,b,c and d are constants. Not sure if this is correct just wanted to double check.
  27. M

    Finding the Second Solution to a Homogeneous Second Order DE

    Homework Statement 4xy'' + 2y' + y = 0 2. The attempt at a solution In class, we were given that y1 = c1Cos(\sqrt{}x). We then used reduction of order to figure out the other solution Yet, I've been trying to figure out, is how do you get y1 in the first place? To me, it doesn't...
  28. M

    Solving the Homogeneous Equation y2dx -x(2x+3y)dy =0

    Homework Statement y2dx -x(2x+3y)dy =0 I have to recognize the equation and solve it Homework Equations The Attempt at a Solution I did y2dx - (2x2+ 3yx) dy=0 which is a homogeneous now after I substitude x=uy dx=udy + ydu I stuck here after the substitution...
  29. H

    ODE Homogeneous Eqn - What did I do wrong this time?

    Hey guys, I've been stuck on this problem for a good hour... I have no idea how to finish it up. Homework Statement Solve: dy/dx = 4y-3x / (2x-y) Use homogeneous equations method. Homework Equations Answer : |y - x| = c|y + 3x|5 (also y = -3x)The Attempt at a Solution dy/dx = 4y-3x / (2x-y)...
  30. H

    ODE: Confused about a Homogeneous Eqn question

    Homework Statement So the problem goes like this: dy/dx = ( x^2 + xy + y^2 ) / x^2 a) Show that it is a homogeneous equation. b) Let v = y/x and express the eqn in x and v c) Solve for y Homework Equations (Included) The Attempt at a Solution a) dy/dx = ( x^2 + xy + y^2 ) / x^2 * [(1/xy)...
  31. A

    Homogeneous Differential Equation

    I read that the 'Homogeneous Differential Equation' is one which has form \frac{\mathrm{d}y}{\mathrm{d}x} = f\left(\frac{y}{x}\right) but I came across one example which was \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x+y}{x+5y} and said that is 'Homogeneous Differential Equation' Too which has 'x'...
  32. C

    Solution of the associated homogeneous problem

    Homework Statement Please help me with this, I would really appreciate it: Using the fact that y1 = x-1/2 cos x is a solution of the associated homogeneous problem, obtain the general solution of x2 y'' + x y' + (x2 - 1/4)y = x3/2 The Attempt at a Solution Well the first thing i...
  33. G

    Isotropic and Homogeneous material

    What's the difference between Isotropic and Homogeneous material? Thanks for your help
  34. C

    Differential Equation - Linear Equations (Non - Homogeneous)

    Homework Statement Find the general solution of \frac{dy}{dt} + 2y = 3t^2 + 2t -1 Homework Equations The Attempt at a Solution So just worrying about the right side y_p = at^2 + bt + c so \frac{dy_p}{dt} + y_p = 2at + b +at^2 + bt + c = 3t^2+2t - 1 at^2 = 3t^2...
  35. H

    Fundamental Solutions of Linear homogeneous equations

    Homework Statement Can y = sin(t^2) be a solution on an interval containing t = 0 of an equation y'' + p(t)y' + q(t)y = 0 with continuous coefficients? Homework Equations The Attempt at a Solution y = sin(t^2) y' = 2tcos(t^2) y'' = 2cos(t^2) - 4t^2sin(t^2) 2cos(t^2) -...
  36. N

    First Order Linear Homogeneous Equation

    First Order Linear Non-Homogeneous Equation Homework Statement I need to solve for e(t) Homework Equations Do I use Laplace Transform for the last integral? The Attempt at a Solution \begin{subequations} \begin{eqnarray} \nonumber \dot{\hat{{\cal E}}}(t) &=& -\kappa...
  37. A

    Second Order Homogeneous Differential Equation.

    I would very much appreciate if anyone can help me with this problem; I have approached it from many different angles to no avail. The position x(t) of a particle moving along the x-axis is governed by the differential equation: x'' + kx' + (n^2)x = 0 , and initially x(0) = a, x'(0) = u...
  38. B

    Is the Universe Homogeneous or Isotropic?

    I recently read an article about whether the http://dailyphysics.com/" and/or isotropic. (the story is at the top - sorry I couldn't get the link to the permanent article to work here) “gargantuan ripples in the density of matter across the universe, known as baryon acoustic oscillations” is...
  39. Z

    Equation w/ Homogeneous Coefficients - y=ux substitution

    Differential Equation w/ Homogeneous Coefficients - y=ux substitution I am teaching myself, this problem is from ODEs by Tenenbaum and Pollard. This is not homework for a class. Homework Statement (x+y){dx} - (x-y){dy} = 0 Homework Equations y=ux, {dy} = u{dx} + x{du} The...
  40. M

    Homogeneous ODE with variable coefficient

    Hello everyone! I'm trying to solve 2 questions for my assignment on homogeneous ODEs I can solve ODE's with variation of parameters and with the method of undetermined coefficients, but these 2 methods seem useless when the coefficients are not constant : (x^2)y" - 2xy' -54y = 0 and...
  41. S

    Solving Homogeneous System of n Linear Equations with Positive Integer k

    Homework Statement Let A * x= 0 be a homogeneous system of n linear equations in n unknowns, that has only the trivial solution. Show that if k is any postive integer, than the system A^k * X = 0 also has only the trivial solution. The Attempt at a Solution I'm so lost please help...
  42. G

    Determine the x-coordinate of the mass center of the homogeneous hemisphere

    Homework Statement Determine the x-coordinate of the mass center of the homogeneous hemisphere with the smaller hemispherical portion removed? I know what the answer should be it's Xcm = 45/112 R Homework Equations The center of mass R of a system of particles is defined as the average...
  43. S

    1st order, non linear, homogeneous, ODE

    Homework Statement "Solve the following ODE's:" "3u+(u+x)u'=0" This is our first weeks homework and he went this through this so quickly in class. Homework Equations None. x and u are both variables. The Attempt at a Solution I know it is homogeneous and non linear. I tried v-substitution...
  44. B

    Why are these DEs homogeneous?

    Homework Statement The following DEs are homogeneous and can be solved using a substitution y=ux or v=xy. I can solve them, I'm just don't see how they are homogeneous. How is it that with there being an exponent (y/x) that this is homogeneous. I don't see how I could pull out t\alpha...
  45. F

    Space-invariant means homogeneous

    The concept of invariance of an object, property, etc... should always expresses with respect to something else: time-invariant means static. space-invariant means homogeneous. direction invariant means isotropic. "Something" can have just one of those types of invariance, all three of...
  46. C

    Homogeneous equation, problem with algebra

    this is the given equation y'=(4y-3x)/(2x-y) and here is all the work I've done so far: (4v-3)/(2-v)=v+x*dv/dx i moved v over and came up with this (-3+2v+v^2)/(2-v)=x*dv/dx did a flip (2-v)/(-3+2v+v^2)dv=dx/x by partial fractions I got a=-3/2 and b=1/2 so...
  47. E

    Homogeneous equation; Initial Value

    Homework Statement Given, (y+2)dx + y(x+4)dy = 0, y(-3) = -1Homework Equations v=y/xThe Attempt at a Solution I've been REALLY struggling with homogeneous equations for some reason...I just don't understand them all. so far I've tried two things. (1)dx -(y)dy ----- ------- (x+4)...
  48. G

    Second Order homogeneous linear differential equation solution

    Hi everyone. I am really confused at the moment learning about Second Order homogeneous linear differential equations. I lay out the background of what I would like to understand. So I understand the actual maths that goes into the diff's, but I do not understand why it should be so, given the...
  49. J

    V substitution in homogeneous equations (diff eq)

    Hey all, i think I'm doing most of this right, but I'm missing a coefficient somewhere when integrating or something... Homework Statement Substitute v=y/x into the following differential equation to show that it is homogeneous, and then solve the differential equation...
  50. G

    Linear System Solutions and the Role of Scalar Multiplication

    I have a question: If the vectors of v & u are solutions of the nonhomogeneous linear system Ax=b, then ru+sv is a solution of the nonhomogeneous system for any real values of r and s. is this statement true? is this statement true for homogeneous systems too?
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