The independent variables X and Y are defined by f(x) = 4ax ; 0 SxSr f (y) 4by; 0 SySs = 0, otherwise = 0, otherwise %3D b-a Show that Cov (U, V) = b + a Where U = x +Y and V =X-Y.

Q: 2 consider, Differentiate with respect to x', dy C」.2ス %3D de

A: From the image , two find a differential equation eliminating the constants from the given relation.…

Q: Let f (x, y) = 3x2 + 2xy + 5y. (a) Calculate df/dx (1, 4). (b) Evaluate df/dy at (x, y) = (1, 4).

A: Differentiate the function f with respect to x treating y as a constant to calculate ∂f∂x.

Q: Determine the Integrating factor of the DE: y (2³e™y – 3) dæ + x (y+ x*e®v) dy = 0 -3 1

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Q: Consider theD. E. M(x, y)dx + N(x, y)dy = 0 if M – N? = 28M, M,N, = 1 and M = 13 – N? then an…

A: Solution:-  Consider  the D.E-               M x,y  dx+Nx,y dy=0         If   My2-Nx3=28M →…

Q: The integrating factor of the DE 2y(x^2 - y + x) dx + (x^2 - 2y) dy = 0 is O x^2 O 2x O e^x O e^(2x)…

A: This question is based on Differential equation.

Q: 1- 2xy 7. y = %3D %3D ds e'-2t co dt 9,

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Q: To reduce the DE with linear coefficients (x+2y – 3) dx – (2x+3y – 4) dy=0 to a homogeneous DE, use…

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Q: Let f(x, y, z) = x² sin y + y e², where x = t + 3,y = net-1 and z In 2t. %3D df at t 1 is: dt %3D…

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Q: Find the integrating function (I. Fn.) for the reducible to exact ODE : (x-y³)dx+xy² dy = 0 а. I.Fn.…

A: Given, x-y3dx+xy2dy = 0  We have to find the integrating function (I.Fn) for the differential…

Q: dw Find dt if w = xy? + x²z+yz², x = t², y = 2t, and z = 2. %3D t=1

A: In this question we have to find the the differentiation of w with respect to w.

Q: When reduced to a linear DE of order one in z, the Bernoulli's equation dy 2 +y √( 1 ) = 3x²y² =…

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Q: Given f(x, y) = 4x²y then the total differential df %3D O 4xy dx + 4x² dy O 8xy dx + 4x² dy None of…

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Q: To evaluate fx (cosx²)e** dx, use integration by parts. What is the best choice for u and du? u = x…

A: Integral by parts.

Q: Find the general solution for dø = z(z – z²)sec²Ø %3D | dz

A: the given differential equation is: dϕdz=zz-z2sec2ϕ we have to find the general solution of the…

Q: There exists a solution, F(x,y)=Dc, to (2- 2aye) da – (2ye + 3 sin y) dy = 0 such that - OF 2ye 3…

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Q: If µ(x, y) = x" y" is an integrating factor of the the DE. (1 + 3xy)dx + (x y-l + x² )dy = 0 Then,…

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

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Q: What is the integrating factor of the linear DE of order one ydx + (3x – 2y+ 5) dy=0? 1 A x² B y3 C…

A: Given linear differential equation is y dx+(3x-2y+5)dy=0 ⇒dxdy+3xy-2+5y=0⇒dxdy+3xy=2-5y Which is…

Q: T "cos(x²)dxdy? What is the reversed order of integration of 'y a. I T cos(x²)dydx ST"cos(x²)dydx O"…

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Q: Let z(x, y) be the function implicitly defined as the solution to TC -2x + y + z + sin(xyz) = %3D…

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Q: Consider theD. E. M(x, y)dx + N(x, y)dy = 0 if M - N? = 24M, M,N, = 1 and M? = 2 - N? then an…

A: We have the differential equation Mx,ydx+Nx,ydy=0 If Nx-MyM=fy then the integrating factor is…

Q: By direct partial integration, solve the following PDE Uxy = Cos (x) cos Ux(x,t) = x %3D and U(T,y)…

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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: By direct partial integration, solve the following PDE Uxy= = cos (x) coS Ux(x,Tt) = x and U(TT,y) =…

A: Given

Q: The PDE ugy + sin(y) u, = cos(y) Can be reduced to a first order linear equation as: A) uz + cos(x)…

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Q: Evaluate //| (x + y – 2z) dV where E = {(x, y, z) | - 2 < y < 0,0 < x < y, 0 < z < x + y²}

A: A detailed solution is given below.

Q: If f(r, y) = 3x2 – 8y, then the total differential of f at z = 1 and y=-2 is: %3D O df = 6 dx + 32…

A: we can find the total differential as below using the properties of partial derivative

Q: Suppose that n # 0 and n # 1. Show that the sub- stitution v = y!-" transforms the Bernoulli…

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Q: To reduce the DE with linear coefficients (X+ 2y - 4) dx – (2x+y-5)dy=0 to a homogeneous DE, use the…

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Q: a When reduced to a linear DE of order one in z, the Bernoulli's equation dx becomes dz A + z( − 1)…

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Q: xyz dV, where E is the rectangular box E = {(x, y, z) : –1 < x < 2,0 < y < 1, –3 < z< 0}.

A: ANSWER:

Q: Given f(x, y) = 8x²y then the total differential df ,2, O 8xy dx + 8x² dy O 16xy dx + 8x² dy O 16x…

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Q: 3. Given: y = f (u) and u = g(x), find dy/dx: a) у%3D сos u, и%3D—х/3 b) y = tan u, u = 10x – 5 - с)…

A: Our aim is to find the derivative of the above expressions.

Q: Find a positive value of k for which y=sin⁡(kt) satisfies d2y/dt2+16y=0.   k=

A: Double derivative

Q: To reduce the DE with linear coefficients (x+ 2y – 3) dx – (2x+3y – 4) dy=0 to a homogeneous DE, use…

A:

Q: 2. Compute ¡ E x dV , where E “ tpx, y, zq | x ď 0, z ď 0, x2 ` y 2 ` z 2 ď 9u

A: The given integral,

Q: dy _Зy +x dx Зx + у

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Q: If y, (x) = e** is a solution of the D.E y" + p(x)y' + q(x)y = 0, where p(x) and q(x) are continuous…

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Q: If f (x, y) = 4e4*cos (y - 1) , then the linear approximation at (0, 1) is %3D (A) 4+ 16x (B) 4+4x…

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Q: dy + ydx %3D = 3x²y?dx,

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Q: The PDE Uxy + sin(y) u = cos(y) Can be reduced to a first order linear equation as: A) Uz + cos(x) u…

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Q: Given z = 3x'y + V5 cos 2y-2x, detemine the total differential o

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Q: The function f(x,y) is homogeneous of degree one. cos a) True b) False.

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Q: flx, v) = { xe-x(1 + y) x20 and y 2 0 otherwise rmine E(XY). ulate Cov(X, Y) and p. X, Y) =

A: Given Two components of a minicomputer have the following joint pdf for their useful lifetimes X and…

Q: If H is the integrating factor of 6xy dx + 5(x² + y)dy= 0, then compute H(8). A 1 в) 2

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Q: Given f(x, y) = - (4x°y + xy® ). Compute: dy?

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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: USE HOMOGENEOUS LINEAR HIGHER ORDER DE 4. y(4) + 50y(²) + 625y = 0 ANSWER: 4. y = (C₁ + C₂x) cos 5x…

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Q: An integrating factor (of the form u(x, y) = x" + y") for the DE. ydx + 5xdy = 0 is O is + y * + y?…

A: Solution: The objective is to find the integrating factor of the given differential equation