Homogeneous Definition and 405 Threads

what does a homogeneous linear equation in 3 variables represent geometrically ?

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|...

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...

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) =...

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 =...

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...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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 ?

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

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

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...

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!

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...

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)...

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?

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...

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] +...

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...

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...

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...

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

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...

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...

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...

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

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...

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) +...

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...

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...

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...

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...

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...

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...

(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 =...

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...

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...

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...

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' =...

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...