# existence

1. ### Smallest interval for existence of unique solution of differential equation

The smallest interval on which a unique solution exist for the IVP yâ€²=e^(2y), y(0)=0, is what? A)|x|<=1/2e B)|x|<=2e C)|x|<=2/e D)|x|<=1/e I am confused which one is correct. Please help.:confused:
2. ### Function Existence

Let a function ð‘“: R â†’ R and some set ð´ âŠ† R be given. Denote by ð‘“ (A) the set of all images that will be obtained by applying the function ð‘“ to the elements of the set ð´. We define a sequence of sets ðºð‘– as follows: a certain subset of real numbers G0 is chosen, and sets...
3. ### Existence of an mâˆˆâ„¤ nâˆˆâ„• such that m/(3^n)âˆˆ(x-r,x+r) xâˆˆâ„

I have shown that the set D={m/(3^n):mâˆˆâ„¤ and nâˆˆâ„•} is countable. I have also shown that there is r>0 such that 1/(3^n) <r for nâˆˆâ„•. My thoughts are: [x-1/(3^n) , x+1/(3^n)]=[((3^n)x -1)/(3^n),((3^n)x +1)/(3^n)]âŠ‚(x-r,x+r). So I need to show that between [(3^n)x...
4. ### Proof: Existence of orthocenter (obtuse triangle)

Hey everybody, I saw a great proof for the orthocenter: https://artofproblemsolving.com/wiki/index.php?title=Orthocenter Now I'd like to show that there exists an orthocenter for obtuse triangles. However, I don't know where or how to start, so maybe you can give me some advice :)...
5. ### existence of sequence with some property

Hello! I have a problem about existence of sequence with some property. Suppose we have monotonically convergence sequence $c_n$ of positive numbers, i.e. $c_n\to 0$, $c_n\geqslant c_{n+1}$ and $c_n\geqslant 0$ $\forall n\in\mathbb N$. Then of course we can find a sequence...
6. ### how do I prove the existence of this norm?

I am reading an article that states: Let $\mathbb{K}$ be a fixed local field. Then there is an integer $q=p^r$, where p is a fixed prime element of $\mathbb{K}$ and r is a positive integer, and a norm $\left | . \right |$ on $\mathbb{K}$ such that for all $x\in \mathbb{K}$ we have \$\left |...
7. ### Prove of existence of another reality beyond our Space-time universe

First of all, let's ask, if our universe is perfect or not. One of forms to see this, is look for the qualities presents in our universe and compare to qualities of a perfect universe, we will see what in our universe the quality of things vary, while in a perfect universe they don't. So, we...
8. ### T(a1) = b1, T(a2) = b2. Show existence of only 1 T

Hello there, Assume : - \vec{a_1}, \, \vec{a_2} \in (\mathbb{R}^2 \setminus \vec{0}) and \vec{a_1},\,\vec{a_2} are not parallel, - \vec{b_1}, \, \vec{b_2} \in \mathbb{R}^2. Show the existence of only one T \, : \, \mathbb{R}^2 \to \mathbb{R}^2 such that T(\vec{a_1}) = \vec{b_1},\...
9. ### Proving the existence of integral

Please, help to prove the following statement: "Let f,g[a,b] \rightarrow \mathbb R be bounded functions, such that f(x) \neq g(x) just in a finite subset of [a,b]. Show that \int _{a}^{b} f(x)dx exists if and only if \int_{a}^{b} g(x)dx exists. Case affirmative, we have \int_{a}^{b} f(x)dx =...
10. ### Existence and uniqueness question

Hi guys! Hope you're all well. A friend and I are struggling with this problem. Consider the initial value problem; 2x + 4y + (4x-2y) (dy/dx) = 0, y(x_0) = y_0 Using the existence and uniqueness criteria, give then region in the x-y place consisting of all points (x_0, y_0) such that there...
11. ### What are the conditions for the existence of x(t) in freq

The Fourier transform of some signal x(t) in continuous time is given by the following equation X(f)=\int_{-\infty}^{+\infty}x(t)e^{-j2\pi f t}dt What are the conditions for the Fourier transform of an arbitrary function or signal x(t) to exist in the frequency domain. Why doesn't x(t) =...
12. ### Existence of the limes

Does this limes exists? Function phi is continuous and bounded. How I can proove existence of the limes? Any ideas are wellcome. [attachment=0:3vzalwnt]PrtScr capture_3.jpg[/attachment:3vzalwnt]
13. ### Existence of limit

A function f is defined by: f(x) = x^4 + 1 if x < 0, 0 if x = 0, x^2 + 1 if x > 0 Find the limit as x-> 0 I know that the one-sided limits as x-> 0+ and x-> 0- are both 1. However, the f(x) = 0 part confuses me a bit. My guess is that the limit does not exist...
14. ### Checking the existence of a solution for a set of equations

I would like to know if there is a method to check the existence of the solution for a given set of linear equations composed with both equality and inequality equations? I'm not interested in the solution, but whether there exist at least one solution or not.
15. ### Establishing existence and uniqueness of solution of an ODE

Consider the initial value problem \frac{dx}{dt} = f(x,t),\;\;x(0)=x_0 If f is continuous with respect to t and locally Lipschitz on x then by standard theorems one can conclude to the existence and uniqueness of a maximal solution on an interval [0,t_f). Now consider the following initial...
16. ### showing existence of nontrivial solution

Show that the differential equation x' = \frac{x \sin(e^x+t)}{1+(e^t \cos x+x)^2} has a nontrivial solution \phi(t) defined on [0,2] such that 0 < \phi(t) < 2 for all t \in [0,2]. I only know that x = 0 is a trivial solution. Then how do I proceed?
17. ### existence of the triangle

a,b,c,d\in R^+ (1) prove there must exist a triangle with three side lengths \sqrt {b^2+c^2}\,\,,\sqrt {a^2+c^2+d^2+2cd}\,\,,\sqrt {a^2+b^2+d^2+2ab}\,\,respectively (2) find the area of the triangle \text Ans:\frac {ac+bc+bd}{2}\text
18. ### existence of a right triangle

(1)for \,\,any\,\,n\in N (2)n>\,2 Prove there must exist a right triangle with all of its three side lengths being positive integers and one of its side length is just equal to n
19. ### Existence of Limits

I am unsure about when exactly a limit exists. I understand that if the right-hand limit is different from the left-hand limit, the limit does not exist. Likewise, if the right-hand limit is the same as the left-hand limit, the limit does exist. However, what if there is only a right-hand...
20. ### show minimum existence

how to show that if f'(a)<0<f'(b),then f must attain its minimum value on [a,b] at a point c in (a,b), assume that f is differentiable can someone tell me how to start???