Uniqueness Definition and 246 Threads

Can someone give me a qualitative example of the uniqueness theorem of a first order linear differential equation? I have read the definition, but I am not 100% positive of what it means in regards to an initial value problem. Im confused about what a unique solution is when/if you change the...

Homework Statement These questions were on my midterm a while ago. I want to understand this concept fully as I'm certain these will appear on my final tomorrow and I didn't do as well as I would've liked on these questions. http://gyazo.com/205b0f7d720abbcc555a5abe64805b62 Homework...

In all of the things that we've discovered have you ever found something as amazing like a magnet? An object that can attract/repel (by using force!) on its own similar other magnets or metals. The only object present to withhold a significant amount (depending on type,size,etc...) of force...

To get the following proof I followed another similar example, but I'm not sure if it's correct. Does this proof properly show existence and uniqueness? Show that if x is a nonzero rational number, then there is a unique rational number y such that xy = 2 Solution: Existence: The nonzero...

Hello, I've seen stated in many places that N=4 SYM is a unique theory and I'm wondering why this is. In my reading, I've seen why there is a unique N=4 supermultiplet and this would fix the field content of the theory (up to one caveat I have: why couldn't an N=4 theory contain multiple N=4...

After studying some of the proofs of equations, worries arises about their uniqueness. 1.Removal of integral ∫∇．Edτ = ∫(ρ/ε)dτ →∇．E = Q/ε 2.Removal of (∇X) ∇XE = ∂B/∂t = ∂(∇XA)/∂t →∇X(E+∂A/∂t)=0 →E+∂A/∂t=-∇V →E=-∇V-∂A/∂t Origin - Certain physics equations/theorems have no...

Homework Statement Show that for any two Dedekind cuts A,B, there exists a unique cut C such that A+C=B 2. The attempt at a solution In order to prove this, I need to prove the existence and uniqueness of such a cut. For the existence, I started by considering a cut for which this works...

The following question seems to be simple enough...Anyway, I hope if someone could confirm what I am thinking. Is canonical transformation in mechanics unique? We know that given \ (q, p)\rightarrow\ (Q, P), \ [q,p] = [Q,P] = constant and Hamilton's equations of motion stay the same in the...

Consider one-sided Laplace transform:$$\mathcal{L} \left \{ h(t) \right \}=\int_{0^-}^{\infty}h(t)e^{-st}dt$$ Q. Is this defined only for the functions of the form f(t)u(t)? If no, then f(t)u(t) and f(t)u(t)+g(t)u(-t-1) are two different functions with the same Laplace transform, and thus...

Hello everyone! So today is was my first day of differential equations and I understood most of it until the very end. My professor started talking about partial derivatives which is Calc 3 at my university. He said Calc 3 wasn't required but was recommend for differential equations. He...

Hello, I'm trying to make a sort of "system theory approach" to dynamic Maxwell's equations for a linear, isotropic, time-invariant, spacely homogeneous medium. The frequency-domain uniqueness theorem states that the solution to an interior electromagnetic problem is unique for a lossy...

Hi! I need some help here, please. In Principles of Mathematical Analysis, Rudin prove the uniqueness of the n-root on the real numbers. For doing so, he prove that both $y^n < 0$ and that $y^n > 0$ lead to a contradiction. In the first part of the proof, he chooses a value $h$ such...

Consider a solid conductor with a cavity inside. Place a charge well inside the cavity. The induced charge on the cavity wall and the compensating charge on the outer surface of the conductor will be distributed in a unique way. How does this follow from the Uniqueness Theorem of EM? David...

there is something i can't seem to get about potential flow, when we work with potential flows we combine some simple potential flows to satisfy some boundary condition (shape of the body and potential at infinity), we get the resulting flow and we assume that that is the flow in reality...

Homework Statement Suppose that T(x, y) satisfies Laplace’s equation in a bounded region D and that ∂T/∂n+ λT = σ(x, y) on ∂D, where ∂D is the boundary of D, ∂T/∂n is the outward normal deriva- tive of T, σ is a given function, and λ is a constant. Prove that there is at most one solution...

Let $f:\mathbb R\to\mathbb R$ be a continuous function of period $2\pi.$ Prove that if $\displaystyle\int_0^{2\pi}f(x)\cos(nx)\,dx=0$ for $n=0,1,\ldots$ and $\displaystyle\int_0^{2\pi}f(x)\sin(nx)\,dx=0$ for $n=1,2,\ldots,$ then $f(x)=0$ for all $x\in\mathbb R.$ I know this has to do with the...

Homework Statement I'm supposed to answer true or false on whether or not the sequence ((-1)^n * n) tends toward both ±∞ Homework Equations Uniqueness of Limits The Attempt at a Solution I did prove it another way, but I would think that uniqueness of limits (as a definition...

Homework Statement We are given (1-z)*f'(z)-3*f(z) = 0, f(0) = 2 valid on the open disk centered at 0 with radius 1 and told to prove there is a unique solution to the differential equation. The hint he gave was to find a factor that makes the left side the derivative of a product, but that...

Ok, after going thru the proof, the only think that still eludes me is the region of definition given in the theorem itself. The rectangle where f and the partial of f with respect to y are known to be continuous. I am thinking of this three dimensionally, and I do not know if this is the...

Ok so ill give an example, x'(t) = log(3t(x(t)-2)) is differential equation where t0 = 3 and x0 = 5 The initial value problem is x(t0) = x0. So what i'de do is plug into initial value problem to get x(3) = 5, so on a graph this plot would be at (5,3)? Then plop conditions into differential...

Here is the proof provided in my textbook that I don't really understand. Suppose that x' and x'' are both limits of (xn). For each ε > 0 there must exist K' such that | xn - x' | < ε/2 for all n ≥ K', and there exists K'' such that | xn - x'' | < ε/2 for all n ≥ K''. We let K be the larger...

Homework Statement x/√(x2+y12)-(l-x)/√((l-x)2+y22)=0 How do I prove that the above equation has a solution for x in ℝ and that the solution is unique? (y1, y2, and l are constants.) Homework Equations x√((l-x)2+y22)-(l-x)√(x2+y12)=0 x√((l-x)2+y22)+x√(x2+y12)=l√(x2+y12)...

Dear MHB members, Suppose that $p,f$ are locally essentially bounded Lebesgue measurable functions and consider the differential equation $x'(t)=p(t)x(t)+f(t)$ almost for all $t\geq t_{0}$, and $x(t_{0})=x_{0}$. By a solution of this equation, we mean a function $x$, which is absolutely...

I'm having trouble understanding what uniqueness is/means. When given a slope/direction field I don't know what I should be looking for if asked to determine if a given initial condition has a unique solution. Example: \textit{y' = }\frac{(x - 1)}{y} With this equation I can see that as long...

Homework Statement This question arises from Chapter ONE, section 2, ex. 7 (d) and (e) of "Introduction to Topology" by Gamelin and Greene. Xbar may be regarded as the completion of a metric space X by identifying each X with the constant sequence {x,x,...}. Show that when Y is the...

Homework Statement http://img267.imageshack.us/img267/8924/screenshot20120118at121.png The Attempt at a SolutionWe have that X = A + B. To show that X is unique, let two such sums be denoted by X1 X2 such that X1 ≠ X2. We write, X1 = A + B X2 = A + B The equations imply, X1 - A - B = 0 X2...

Homework Statement http://img854.imageshack.us/img854/5683/screenshot20120116at401.png The Attempt at a SolutionSo we have that A + B is a vector in S + T, where A is an element of S and B is an element of T. Suppose there is another vector A' + B' also in S + T, where A' is an element of S...

Homework Statement http://img854.imageshack.us/img854/5683/screenshot20120116at401.png The Attempt at a SolutionSo we have that A + B is a vector in S + T, where A is an element of S and B is an element of T. Suppose there is another vector A' + B' also in S + T, where A' is an element of S...

FYI this is a homework problem which I already have the answer to but would like to clarify some points on. Homework Statement Find the Smith Normal Form of the matrix \left[ \begin{array}{cccc} 6 & 0 & 4 \\ 0 & 6 & 8 \\ 0 & 3 & 0 \end{array} \right] over the ring of integers. Homework...

Hi everyone, I'm not quite sure how to proceed to show existence (and perhaps uniqueness) of the following system of (first order) differential equations: \dot{x}=f(t_1,x,y,z) \dot{y}=g(t_2,x,y,z) \dot{z}=h(t_3,x,y,z) where \dot{x}=\frac{\partial x}{\partial t_1}...

How would you show mathematically that Newton's laws, when taken as given, always yield a motion and that this motion is always unique (given initial positions/velocities) for arbitrary systems?

Hi all. Suppose that U1 is the solution of the Laplace's equation for a given set of boundary conditions and U2 is the the solution for the same set plus one extra boundary condition. Thus U2 satisfies the Laplace's equation and the boundary conditions of the first problem, so it's a solution...

Homework Statement S and T are two affine lines in \mathbb{A}^3 that are not parallel and S\cap T=\emptyset. Show there is a unique affine plane R that contains S and is weak parallel with T. The Attempt at a Solution Existence is easy, if S=p+V and T=q+W then R=p+(V+W) satisfies the...

Homework Statement If x > 0, then there exists a unique y > 0 such that y2 = x. The attempt at a solution Proof. Let A = {y ∈ Q : y2 < x}. A is bounded above by x, so lub(A) = η exists. Suppose η2 > x, where η = lub(A). Consider (η - 1/n)2 = η2 - 2η/n +1/n2 > η2 - 2η/n. Now η2 -...

I'm having difficulty understanding this concept of uniqueness. What's the precise definition of it? Let say we have some direct sum decomposition, (1)Are \left( {\begin{array}{*{20}{c}} {{R_1}} & 0 \\ 0 & {{R_2}} \\ \end{array}} \right) and \left( {\begin{array}{*{20}{c}}...

My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x0, then the initial value problem dy/dx + P(x)y = Q(x), y(x0)=y0 has a unique solution y(x) on I, given by the formula y=1/I(x)\intI(x)Q(x)dx where I(x) is the...

Homework Statement Solve the Cauchy problem: (t2 + 1)y' + etsin(t) y = sin(t) t2 y(0) = 0 Homework Equations y'(t,y) + p(t)y = g(t,y) Integrating factor e(integral of p(t)) The Attempt at a Solution I tried finding an integrating factor, but it came out ugly. I couldn't solve the...

Hello, I have a PDE: 3*u_x + 2*u_y = 0, and I am interested in determining initial values such that there is a unique solution, there are multiple solutions, and there are no solutions at all. What theorem(s)/techniques would be of use to me for something like this? Regards, Dan

Is my proof correct? Homework Statement Show that, if there exists a number 0 for which x+0=x for all x∈R, and a number 0' for which x+0'=x for all x∈R, then 0=0'. The Attempt at a Solution Proof by contradiction: Assume, 0≠0'. Then, x+0=x and x+0'=x' Such that: x=x-0 and x=x'-0'...

Homework Statement Show that the solution of dx/dt=f(x), x(0)=xo, f in C^1(R), is unique Homework Equations C^1(R) is the set of all functions whose first derivative is continous. F(x)=integral from xo and x (dy/f(y)) The Attempt at a Solution Assume phi1(x) and phi2(x) are both...

Hi, Just wanted to ask a question regarding existence and uniqueness of solutions to SDEs. Say you have shown existence and uniqueness of a solution to an SDE that the process [tex ] X_{t} [/tex ]a particular process follows (by showing drift and diffusion coefficients are Lipschitz). If you...

Homework Statement Check if the given initial value problem has a unique solution Homework Equations y'=y^(1/2), y(4)=0 The Attempt at a Solution f=y^(1/2) and its partial derivative 1/2(root of y) are continuous except where y<=0. We can take any rectangle R containing the...

Homework Statement Check if the given initial value is a unique solution. Homework Equations y'=y^(1/2), y(4)=0 The Attempt at a Solution I got y(t)=(t/2)^2 and 0 at t=4 So, we have two solutions to i.v.p.; therefore, it's not a unique solution. Is it correct?

Hello! I would like to prove the following statement: Assume f\in C^{1}(\mathbb{R}). Then the initial value problem \dot{x} = f(x),\quad x(0) = x_{0} has a unique solution, on any interval on which a solution may be defined. I haven't been able to come up with a proof myself, but would...

(Hopefully, this question falls under analysis. I was unable to match it well with any of the forums.) The proof that the identity element of a binary operation, f: X x X \rightarrow X, is unique is simple and quite convincing: for any e and e' belonging to X, e=f(e,e')=f(e',e)=e'. However...

As far as I can understand it, Picard's Existence and Uniqueness of ODEs theorem relies on the fact that a the given function f(x,t) in the initial value problem dx/dt = f(x,t) x(t0) = x0 is Lipschitz continuous and bounded on a rectangular region of the plane that it's defined on. And the...

Homework Statement Consider the IVP compromising the ODE. dy/dx = sin(y) subject to the initial condition y(X) = Y Without solving the problem, decide if this initial value problem is guaranteed to have a unique solution. If it does, determine whether the existence of that solution is...

Homework Statement Two systems are given (both are completely controllable): x-dot = Ax + bu z-dot = A*z + b*u They are related by the state transformation: z=Tx prove that the transformation matrix T is unique. The Attempt at a Solution Since the systems are completely controllable, we...

Homework Statement for the differential equation t^2y''-2ty'+2y=0 with the general solutions y=C(t) + D(t^2) where C and D are constants. given the inital solution y(0)=1 and y'(0)=1 there are no solutions that exist. Why does this not contradict the Existence and Uniqueness Theorem...

I know this probably sounds weird, but I have a research problem that requires "random" analog circuits. Basically what this means is that I create Spice netlists by randomly adding linear and/or nonlinear components of random types with random node and parameter values. This works fine and I...