Homogeneous Definition and 382 Threads

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=y[SIZE="2"]1(x) is a solution and k is a constant, then y=ky[SIZE="2"]1x is also...

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...

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...

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 \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...

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...

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...

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...

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?

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...

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...

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?

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...

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...

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...

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...

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...

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...

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=...

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

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!

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...

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...

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...

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

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...