y'(t) = f(t, y), t ε io, 1], y(0) = 1. For eaching of the following f(t, y), find out whether f is Lipschitz continuous in the variable y on the set D {(t,y)10 t 1,-oo < y < oo), and de...
10. (a) 5pts] Prove that if f is piecewise continuous on [0,00) and satisfies the growth is, f(t) Me for restriction that some constants M k>0 on [0,00), then lim f(t)]=0 (b) [5pts] Find the inverse transform of s+1 F(6) s) = ln (c) [8pts] Find a solution of ty"+2y-ty= 0, y(0)=1, y (1)= Hint) You may need to use (a) and (b). 10. (a) 5pts] Prove that if f is piecewise continuous on [0,00) and satisfies the growth is,...
2) (a)(10 pts.) Find the continuous solution to the initial value problem de + y = 9(2) where q() = { 0 if 2>1 sat S 1 if |2<1. satisfying y(0) = 0. (b)(10 pts.)Solve the differential equation de ty
3. Draw the direction field of the following differential equation: = (1-y)y dt What happens for the solution satisfying y(0)-2, 1, 0.5,-1 as t-> oo? If y(2)-β and limt→oo y(t) = 1. Find all possible values of β. 3. Draw the direction field of the following differential equation: = (1-y)y dt What happens for the solution satisfying y(0)-2, 1, 0.5,-1 as t-> oo? If y(2)-β and limt→oo y(t) = 1. Find all possible values of β.
1. Let X be a continuous random variable with CDF F(ro)-a+b 3 and support set 0, 1]. (a) Calculate the values of a, b that would make F(ro) a valid CDF. (b) Write out the pdf of X. c) Calculate EX d) Calculate EX
Consider the following initial value problem. y′ + 5y = { 0 t ≤ 1 10 1 ≤ t < 6 0 6 ≤ t < ∞ y(0) = 4 (a) Find the Laplace transform of the right hand side of the above differential equation. (b) Let y(t) denote the solution to the above differential equation, and let Y((s) denote the Laplace transform of y(t). Find Y(s). (c) By taking the inverse Laplace transform of your answer to (b), the...
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then lim f(x)- lim F(x) "( T / F ) If lim -f(x)s lim. f(x) L, then lim f(x)s 1. "(T/F) lim. In x -oo . (T/F) lim0 ·(T / F ) The derivative f' (a) is the instantaneous rate of change of y...
Find the solution of the initial value problem y" – 2y' + 5y = g(t), y(0) = 0, y'(0) = 0, where g(t) is a continuous, otherwise arbitrary, function. Oy(t) = g(t) 1 y(t) = (sets sin(2t))g(t) Oy(t) = (cos(2t)) * g(t) Oy(t) = (cos(2t))g(t) y(t) = (1 e*) + f(t) x(t) =() e sin(26)g(t) g(t) = ( e sin(2t) + (t) y(t) = Ce+ sin(2t)) *g(t) 1
Let X be a continuous random variable with the following probability density function f 0 < x < 1 otherwise 0 Let Y = 10 X: (give answer to two places past decimal) 1. Find the median (50th percentile) of Y. Submit an answer Tries 0/99 2. Compute p (Y' <1). Submit an answer Tries 0/99 3. Compute E (X). 0.60 Submit an answer Answer Submitted: Your final submission will be graded after the due date. Tries 1/99 Previous attempts...
Let f : R2 → R be a uniformly continuous function and assume that If(y,t)| M. Let yo E R. The goal of this exercise is to show the existence of a function φ : [0, 1] → R that solves the initial value problem o'(t)-F(d(t),t), ф(0)-Yo (a) Show that there is a function n1,R that satisfies t <0 n(リーレ0+.GF(du(s-1/n),s)ds, t20. Hint: Define фп first on [-1,0] , then define фп。n [0,1 /n), then on [1/n, 2/n], and so on...
1. Use combinations of STEP FUNCTIONS to describe each continuous-time signal shown below. f(t) 0 2 4 6 0 1 2 3 0 1 2 3 4 2. Sketch the following signals: (a) x (t)=1 [u(t+2)-u(t-1)] (c) X(t)=\fety (b) X(t)=t.e (d) x (t) = u(t) u(t-1).ult-2).u(t-3) 3. Determine whether the systems below are linear and time invariant. Justify your answer! (a) y(t) = x(31) (b) y(t)= 2x(1-t) y(t)=cos(x(t)] 4. Simplify the expressions: (a) y(t)=1.8(t+2)+(t +1) 8(1-1)+(t+3). 8(t) (b) y(t) =...