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 < ∞ Find the Cov(Y1, Y2). Are Y and Y2 independent?

Q: 3. Find the maximum and minimum values of a 1 āz - on |z| ≤ 1 (where a with la| < 1 is fixed)

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Q: A second order Caushy – Euler equation having a solution y(x) = x-cos(4Inx) is O x²y" + 5xy' + 16y =…

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Q: Without solving for y(x), find the value of y" (0) in the initial value (a) problem (x + 2)y' + y =…

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Q: 59. Prove the general formula iwollo oi ons dx 1 x - In x – b + C %3D (x – a)(x – b) a – b - - where…

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Q: 4.1 In aerodynamics and fluid mechanics, the functions u(x,y) and v(x,y) in f(z) = u(x,y) + iv(x,y),…

A: As per the policy, we are solving the first question. Please repost the question and specify which…

Q: 6. Suppose that the coefficient functions p(t) and q(t) are continuous in the interval (0, x), and…

A: Use the formula of Wronskian to determine its values. The functions are already given. Determine…

Q: 29. In Theorem 9.8, if f'(x o) = 0, no conclusion can be drawn about the equilibrium point xo of x'…

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Q: 1 10-10z Set up the iterated double integral f S S dy dr as dxdy such that

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Q: full solutions please, thank you

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Q: 4.Determine the correct Wronskian setup for v, and v, for the equation y" – 3y + 2y = 10sin(x) e 2x…

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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: 5. Determine a function v(r, 0, ø) with Av = 0 _in B2 \ B1 satisfying the boundary conditions v(1,0,…

A: Use the cylindrical coordinates to determine the value of r. Now use the boundary conditions to…

Q: If u(x, t) = D1 fn(t) sin(nx) is the solution Utt – 9ura = 2e2t sin(3x) + e-2t sin(4x) u(0, t) = 0 =…

A: Given: ux,t=∑n=1∞fntsinnx is the solution of the PDE…

Q: The governing PDE: y =cy, 0<x<L subject to the BCs: XX dy = 0, y (L, t) = 0 dxlx=0 and the ICs: dy…

A: Given condition does not satisfy the solution. Please check it first.

Q: 6. Suppose that the coefficient functions p(t) and q(t) are continuous in the interval (0, x), and…

A: Given differential equation is y''+pty'+qty=0 Given y1t=t and y2t=sint are the solutions of the…

Q: 4. Solve the Cauchy problem Utt = 9uxx+cos x, (x, t) E Rx (0, ∞) u(x,0)=0, TER u(a,0)=0, x ER using…

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Q: Given that y1 = zi cos(z) and y2 = x¯i sin(z) form a fundamental set of solutions of z*y" + zy' + (z…

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Q: If ulr. 0) satisfies V'u = 0 inr<1, u(1, 0) = S(e). show that Poisson's integral formula in R' takes…

A: Given, ur,θ=0 satisfies ∇2u=0 in r&lt;1. Also u(1,θ)=fθ To show ur,θ=1-r22π∫-ππfϕ1-2rcos(θ-ϕ)+r2dϕ…

Q: Consider h(x,y, z) = cos (xy) + eY² + In (xz) at the point P ( 1,0, = 1 Deter-

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

A: Using diffenrentiation

Q: 9. Given z=x siny + 2x*y- cos(xy), calculate or z yr C) cosy+4x +x cos(xy) + ysin(xy) D) cosy+4x +…

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Q: f(x) = In(2 + cos x). -2.125, xk+1 = Ik +0.5, yk = f (xk), k E N To = %3D %3D = 0r

A: The function is fx=ln2+cosx Conditions: x0=-2.125, xk+1=xk+0.5, yk=fxk, k∈ℕ xk+1=xk+0.5…

Q: If y is a solution of y(x)-[(x-1t)»(?)dt = 1, then which of the following is true? (a) y is bounded…

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Q: 2) If g(x) = dt then g'(x) • (1+ 2t)' 3 -1 (a) (1 + 2x)* * (b) +e (d) (1+2x)ž +e (1+ 2x)³ (1+2x)à *…

A: As per our guidelines, we are allowed to answer the first question only please resubmit other…

Q: e) Suppose that W= √√x² + y² + z², x= cos 0, y = sin 0, z = tan 0. dw T Use the chain rule to find…

A: According to our company's guidelines I can only answer (e), since (e) and (f) are not inter…

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: 6. Suppose that the coefficient functions p(t) and q(t) are continuous in the interval (0, t), and…

A: if you find my answer helpful please put thumbs up. thank you. the required solution is given below…

Q: Integrate the fuackion f=g regian bonndel by y the en6* y-o ver

A: The function is fx=xyThe region is bounded by y=x, y=6-x, y=0

Q: 6. Suppose f(2) is analytic in |2| 1, and f(0) = 1, f(0) = 2, compute: %3D (1) h = fe(s+ 1)2d: (2)…

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Q: find c, and c2 so that y(x) = c, sin x+ c2 cos x will satisfy the given conditions. Determine…

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Q: t Consider the DE 'y" - zy' - 8y = 3z - A) Verify that y =and y =z satisfy the DE: z'y" - zy -8y =…

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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: 1. Let z = f(x, y), find for: and dy - - (a) e"y + z ln y = 0 (b) yz – sin z = xy

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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: Suppose that ƒ € H(C) and f(n) = 0 for all n e Z. Show that all the singularities of f(2)/sin(rz)…

A: Given that fz is an entire function with fn=0 ∀n∈Z and let gz=sinπz We have to show that fzgz has…

Q: 15.If the Wronskian of f and g is t cos t − sin t, and if u = f + 3g, v = f − g, find the Wronskian…

A: Consider the provided information, Wf,g=fg'-gf'⇒tcost-sint ...... (1) Find the Wronskian of u and v.…

Q: The solution of the nonhomogeneous DE y" – 5y' +4y= 4x+3 is y=x-2 y= cje*+ c,e* + x- 2 y = c;e +…

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Q: If J f(x) cos(mx)dx = and f(x) = En=1 cos(n²z) n3+1 then m = O 4 O 25 O 16 0 4 9 O 5 O2 03

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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: 3. Let V = eP cos(qr), with p = x In(xyz), q = z cos(Ty) and r = 2 tan-(y). %3D By using the chain…

A: Solve the given problem.

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: 2. Let exp(z) = 2 + i find z in x + iy form. 3. Find the solution of In(z) = – T.

A: use: ln(x+iy)=ln|x+iy| + arg(x+iy)     where, arg(x+iy)=tan-1(y/x)

Q: Let (x² – x)ʻy" +y' -y=0,the D.E has regular singular point at x =

A: The DE x2-x2y''+y'-y=0. Then we have to find the regular singular points for the given DE.

Q: Q1) Suppose w =÷+ ?, x = cos² t, y = sin² t, z = ; Evaluate dw at t = 3, by using the Chain Rule dt

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Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.