In Exercise 5.8, we derived the fact that 4y1 Y2. 0< yı < 1,0 < y2 < 1, f(y1. y2) = 0. elsewhere. Show that Cov(Y1, Y2) = 0. Does it surprise you that Cov(Y1, Y2) is zero? Why?

Q: Solve the solution of the DE of the Gompertz function: dP = c In[P dt М In[P

A: We have a nonlinear ordinary differential equation :

Q: a. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian. b. Establish…

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Q: Calculate the energy coefficient from the first order if the infinite potential box is as uneasy as…

A: Given: φn=2LsinnπxLφ2=2∂Lsin2πx2L To find: Solution

Q: The homogenous D. E. with constant coefficient of at least order which has the solution y = 2x? +…

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Q: Suppose E(X) = 1, V(X) = 2, E(Y)= 3, V(Y) = 4, and E(XY)= 5. (1) Сov (X, Y) (2) Corr(X,Y) (3)…

A: 1) COV(X,Y) =E(XY)-E(X)E(Y)                    = 5-1×3                    = 2

Q: 1. Using the first prine =) , show that (1 – x*) + 4nxy = 0. %3D 2. If y = 1+x2, + qx2+rx +s passes…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: In (T/e) ( (2- sec?x)eme sec³x sec (зА - х) -dx to be integrable using the formula и" du? ? What…

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Q: Q3: show that: S yi va -y dy =na² where a is a positive constatnt

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Q: In the integration of (csc 3t)(cot 3t) dt, what is t O cot 3t none of the above O csc 3t O (csc…

A: See the details solution in below

Q: The integral of the given function ∫csc^6(x) dx is equal to

A: Need to find ∫cosec6(x) dx

Q: 2(3) Calculate the following commutators: (a) [d/dx, x·d/dx] (b) [sin(x), cos(x)] (c) [x^2, Ĥ] ,…

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Q: Use Secant method to to approximate the solution of f(r) = z+ cos z =0 accurate to within 10-3,…

A: The secant method is used to calculate the root of the function. The formula for the secant method…

Q: 1 prove that | e -ym dy = mr(m) -ут

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Q: Find the value of dr/dt at t = 0 if r = (u2 + 7)^(1/3) and u^2t + u = 1.

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Q: ppose X1,...,Xn are i.i.d.U(0,θ). You want to test H0: θ=θ0 versus H1:θ < θ0 at level α? Derive the…

A: The given information is: Xi~U0, θ or fx=1θ,   0&lt;x&lt;θ The null and alternative hypotheses are:…

Q: Consider the non-homogeneous O DE y +2 y +2y = sin 2x using undetermined coefficients method, the…

A: Given differential equation is y''+2y'+2y=sin2x To find the Particular solution yp of the…

Q: The 300-lb block shown in the figure is kept from sliding down the inclined plane by friction and a…

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Q: 4. Find the Cauchy principal values of the following integral: dx. (x²+1)(x²+2x+2)

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Q: Evaluate the integral. 5n /2 2, 1 - cos t dt 3π/2

A: ∫3π25π21-cos2tdt=∫3π25π2sin2tdt=∫3π25π2sintdt=∫3π22π-sintdt+∫2π5π2sintdt=1+1=2

Q: solve the given nonhomogeneous ODE x^2y"-2xy'+2y=x^3sinx by varition of parametrs

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Q: Evaluate the integral. 1 cos 2t dt 2 7x/2

A: To evaluate the given integral.

Q: Find the linear approximation of fcn)cn" In n around n=e Using this equation, y = f(a)+f'ca)(n-a)

A: Applying linear approximation we have

Q: In Exercises 1-8, find the general antiderivative of f and check youranswer by differentiating. (x)…

A: We have to find antiderivative and check the solution for the given function: fx=4x7-3 cosx Finding…

Q: The auxiliary (charasteristic) equation of the DE x²y" + xy' + y = 0 is %3D

A: Euler-Cauchy type differential equation: Let a equation x2y''+y=0 ..(i) Put y=xm in the given…

Q: Find the integration inner product on C[–4, 1] of the given functions. f(x) = 15x g(z) = 3r* (f, 9)…

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Q: Consider the following integral: +oo J(2a) = | - dzerp(-20r") / dyezp(-2ay7) = { / drerp(-2aa")}?…

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Q: 2. Evaluate the integral 8- (a) L« (4t5/2 – 3t3/2) dt (d) 2 - Vt dt r4 (e) t2 cos e de JT/4

A: The objective is to find the values of given integrals

Q: Compute the value of the following integral: ∞ (t) cos() 8 (t-1)dt ∞ 2 where 8(t) is the unit…

A: According to the definition ∫f(x)dx=F(x)+C, where dFdx=f(x). A delta function is defined as δ(t-a)=0…

Q: Suppose that the joint p.m.f. of Y and Y2 is defined by: 2x + y for x = 1, 2,3; y = 1,2,3 54 f(x, y)…

A: The probability mass function of Y1 and Y2 is,

Q: 20 Given that the (bx) +(ay) =(ab), where a and b are constants. Show that-

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Q: 7. a) Prove that for r > 0 * (E) + c. Vr2 – r²dx = vr² – x² + -1 sin 2 +C.

A: To prove: ∫r2-x2dx=x2r2-x2+r22sin-1xr+C

Q: 26.Show that if p is differentiable and p(t) &gt; 0, then the Wronskian W(t) of two solutions of…

A: Given: [p(t) y′]′ + q(t) y = 0 W(t) = c/ p(t)

Q: Use the I.V.T and Rolle's Theorem/M.V.T to prove that the equation below has a unique solution in…

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Q: Consider the S.mpto lincar rogressic 3:0n m adol Y tX; t U ukon exp(az.) You have the fallow. ng…

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer three sub-parts at a…

Q: 2. Evaluate ř(t) dt: 4 4t ř(t) t2 + 5t + 6' î + 2e4t cos t j - k; a = 0 and b = 4 t2 + 1

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Q: Find the genercal homogeneous second linear ditferential given solution of the order s+equatio.…

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Q: Let {yk } be the DFT of {x; }, where x = [0, –1, 0, 1] y3 = O i -1 -i 1

A: Here, x0=0, x1=−1, x2=0 and x3=1 We have to find y3 Now, y3 is given by…

Q: Let f1(x, y) - e cos(5y), f2(x, y) = x² – 25 y². Find Vf, and Vf2, and show that %3D (fv) = 0 for…

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Q: 2. Derive a formula to approximate S* s* f(x;y)dxdy yo xo Using Simpson's rule with x2 – xo = 2h ,…

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Q: _Suppose w =+, x = cos² t, y = sin? t, z = ; dw Evaluate at t = 3, by using the Chain Rule dt

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Q: Find the value of Res(f'(2), 20) if zo is an isolated singularity off.

A: Let us start finding the value of  Res(f'(z),z0)   z0 is an isolated singularity of f

Q: show that f(z)=(z-1) cos(1/z+2) has an essential singularity at z=-2

A: Introduction: Laurent series has an important role in determining the singularities. Suppose, a…

Q: Apply the techniques of Example 7.1.5 to the evaluation of the improper integral 8 dx 1 = (a) Let…

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Q: 1.Suppose that the joint p.d.f. of Y, and Y, is defined by: e-Y1/y2 · e-Y2 0 < y1 < 0, Y2 0 < y2 < ∞…

A: Given: The joint p.d.f of Y1 and Y2 is defined by:

Q: that X and Y have joint pdf f(x, y) ={x + 3x^2y if 0 <= x <= 1 and 0 <= y <= 1, 0 otherwise.   (c)…

A: From the given information, Consider, the Joint pdf of X and Y is as follows: fx,y={x+3x2y if…

Q: Suppose w = In(æ² + y² + z²) , Compute dw/du and dw/dv in two different ways: ue3" sin(9v) , y :…

A: GIvenw=ln(x2+y2+z2) where x=ue3vsin(9v) , y=ue3vcos9v and z=ue3vTo find∂w∂u and ∂w∂v

Q: If (х, у) %3D х" у"is an integratingfactor of the the DE. (1+ 3ху)dx + (ху-1 +x?)dy %3 0 Then, the…

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Q: 1- Find dwi dt at the given value for w ==+2,x = cos (1), y = sin (1),z = w =-+ at = 3

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Q: Suppose w=x/y+y/z, where x=e^(4t), y=2+sin⁡(4t), and z=2+cos⁡(3t). A ) Use the chain rule to find…

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Q: Let u = cos(4x) and du = e² da, then by using IBP, the integral [e²= cos(4x) e2x cos(4x) dx is equal…

A: Given, u=cos4x and dv=e2xdx. Formula for IBP: ∫udv=uv-∫vdu

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.