For Exercises 1-4 below, (a) verify that y1 and y2 satisfy the given second-order equation, and...

For Exercises 1-4 below, (a) verify that y1 and y2 satisfy the given second-order equation, and (b) find the solution satisfying the given initial conditions (I.C.).

2. y′′−3y′+2y=0; y1(x)=e^x,y2(x)=e^2x. I.C.y(0)=0,y′(0)=−1.
3. y′′−2y′+y=0; y1(x)=e^x,y2(x)=xe^x. I.C.y(0)=1,y′(0)=3.

Expert Solution

Colby Messinger answered 1 year ago

a) Verify that y1 and y2 are fundamental solutions of the given homogenous second-order linear differential...

a) Verify that y1 and y2 are fundamental solutions of the given homogenous second-order linear differential equation b) find the general solution for the given differential equation c) find a particular solution that satisfies the specified initial conditions for the given differential equation y'' - y = 0 y1 = e^x, y2 = e^-x : y(0) = 0, y'(0) = 5

The joint density of Y1, Y2 is given by f(y) = k, −1 ≤ y1 ≤...

The joint density of Y1, Y2 is given by f(y) = k, −1 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1, y1 + y2 ≤ 1, y1 − y2 ≥ −1, 0, otherwise a. Find the value of k that makes this a probability density function. b. Find the probabilities P(Y2 ≤ 1/2) and P(Y1 ≥ −1/2, Y2 ≤ 1/2 c. Find the marginal distributions of Y1 and of Y2. d. Determine if Y1 and Y2 are independent e....

Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1 0, otherwise. a)...

Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1 0, otherwise. a) Are Y1 and Y2 independent? Why? b) Find Cov(Y1, Y2). c) Find V(Y1−Y2). d) Find Var(Y1|Y2=y2).

a) verify that y1 and y2 are fundamental solutions b) find the general solution for the...

a) verify that y1 and y2 are fundamental solutions b) find the general solution for the given differential equation c) find a particular solution that satisfies the specified initial conditions for the given differential equation 1. y'' + y' = 0; y1 = 1 y2 = e^-x; y(0) = -2 y'(0) = 8 2. x^2y'' - xy' + y = 0; y1 = x y2 = xlnx; y(1) = 7 y'(1) = 2

In exercises 1–4, verify that the given formula is a solution to the initial value problem....

In exercises 1–4, verify that the given formula is a solution to the initial value problem. 2. Powers of t. b) y ′ = t^3 , y(0) = 5: y(t) = (1/5)t^(4) + 5 3. Sines and cosines. a) x′ = −y, y′ = x, x(0) = 1, y(0) = 0: x(t) = cost, y(t) = sint

Suppose that Y1 and Y2 are random variables with joint pdf given by f(y1,y2) = ky1y2...

Suppose that Y1 and Y2 are random variables with joint pdf given by f(y1,y2) = ky1y2 ; 0 < y1 <y2 <1, where k is a constant equal to 8. a) Find the conditional expected value and variance of Y1 given Y2=y2. b) Are Y1 and Y2 independent? Justify your answer. c) Find the covariance and correlation between Y1 and Y2. d) Find the expected value and variance of Y1+Y2.

Find the general solution of the given second-order differential equation. 1. 4?'' + 9? = 15...

Find the general solution of the given second-order differential equation. 1. 4?'' + 9? = 15 2. (1/4) ?'' + ?' + ? = ?2 − 3x Solve the differential equation by variation of parameters. 3. ?'' + ? = sin(x)

Use the method of reduction of order to find a second solution y2 of the given...

Use the method of reduction of order to find a second solution y2 of the given differential equation such that {y1, y2} is a fundamental set of solutions on the given interval. t2y′′ +2ty′ −2y=0, t > 0, y1(t)=t (a) Verify that the two solutions that you have obtained are linearly independent. (b) Let y(1) = y0, y′(1) = v0. Solve the initial value problem. What is the longest interval on which the initial value problem is certain to have...

1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|. (a) Prove...

1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|. (a) Prove that (R2,ρ) is a metric space. (b) In (R2,ρ), sketch the open ball with center (0,0) and radius 1. 2. Let {xn} be a sequence in a metric space (X,ρ). Prove that if xn → a and xn → b for some a,b ∈ X, then a = b. 3. (Optional) Let (C[a,b],ρ) be the metric space discussed in example 10.6 on page 344...

For the following exercises, graph the inequality. 1/4 x2 + y2 < 4

For the following exercises, graph the inequality.1/4 x2 + y2 < 4

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