(0) 4 P10ỤI D UOITUIIILLIOII. 9. Prove that there exists a unique solution to the equation...
3. Consider the equation: xy' + y² +y=0 (a) Show that a solution exists and is unique for all initial conditions of the form y(a) = b where 0 < a < 10 and 0 < b < 10.
4. Consider the differential equation with initial condition r(0) = 0 (a) What does the existence and uniqueness theorem tell you about the solution to this IVP? (10 points) (b) Use separation of variables to find the solution for the IVP r(to) = Io for to +0. (5 points) (c) Are the solutions to b) unique? (5 points) (d) Sketch solutions for Xo = --1,0,1 and to = 1 and show that for all to and to the solution goes...
Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. (25 − y2)y' = x2 Choose the right answer and explain a. A unique solution exists in the regions y < −5, −5 < y < 5, and y > 5. b. A unique solution exists in the region y < 5. c. A unique solution exists in the region consisting of...
(2) Prove that if j-0 i-0 with k, 1 e N u {0), and bo, . . . , be , do, . . . , dl e { 0, . . . , 9), such that be, de # 0, then k = 1 and bi- di fori 0,.. , k. (I recommend using strong induction and uniqueness of the expression n=10 . a + r with a e Z and re(0, 1, ,9).) (3) Prove that for all...
Suppose U is a set. Prove that there exists a unique set A E P(U), such that An B = A for any BE P(U).
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such that f(0) 9(0)-1. Show that there exists some δ > 0 such that ifTE 0,d) then (b) Consider the function 0 l if z e R is rational, if zER is irrational f(z) Show that limfr) does not exists for any ceR. 4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such...
I have first part of question good. Need to prove unique modulo and do not know where to start. Prove that the congruences x-a mod n and x b mod m admit a simultaneous solution if and only if (n, m) | (a -b). Moreover, if a solution exists, then the solution is unique modulo [m, n). Prove that the congruences x-a mod n and x b mod m admit a simultaneous solution if and only if (n, m) |...
9. (4 points) Does there exist a unique solution to the following IVP in a neighborhood of the initial condition? Find all constant solutions, if any. Specify the largest interval over which your constant solution is valid. Justify your answers. dy (ey-)(tan y) In(1 -),y(0)-3/2 9. (4 points) Does there exist a unique solution to the following IVP in a neighborhood of the initial condition? Find all constant solutions, if any. Specify the largest interval over which your constant solution...
For D.5 through D.11, find the solution set using Gauss elimination, if a solution set exists. Also, classify "inconsistent", "consistent with a unique solution", or "consistent with an infinite number of solutions" D.5. 10
11 > 0, then there exists a subset of A that is not Prove that if A CR and A Lebesgue measurable.