A function f(x, y) = x3 − 3xy2 + y3 is homogeneous of degree n when                f (tx, ty) = tnf (x, y). (a) show that the function is homogeneous and determine n, and      (b) show that xfx(x, y) + yfy(x, y) = nf (x, y).

Q: 14. Show that the function f: R²R given by if (x, y) 0, ²+y² f(x, y) 0 if (x, y) = (0,0), is…

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Q: 3.2= e'-ry ; I=t3 and y = t° %3D

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Q: Determine the type and stability of the critical points of the homogeneous systems: dx = x - 2y dt…

A: We have to determine the type and stability of the critical points of the homogeneous systems.

Q: Find the first partial derivatives of (a) f(x,y,z)=x²yz³ (c) S(x,y,z)= In(#)+ tan""(yz) (b)…

A: Solution of first partial derivatives.

Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 4x²y - 5xyz + 7yz?…

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Q: Show that f(x, y) = xey + ye" is a solution of the equation fraz (x, y) + fyy (x, y) = x fryy(X, Y)…

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Q: 3. Find the total differential of z = f(x, y) if z2 - y = (1– 2x)z.

A: Total differential of a function z=fx,y is dz=∂z∂xdx+∂z∂ydy

Q: a. w = √√x² + y² + z² xy b. w = x+y+z c. w = In√√x² + y² + z² 1

A: Use the procedure of partial derivatives to find with respect to x y and z as in next step.

Q: y" - 2y'- 8y = 2xe* %3D

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Q: A. Find all the first order partial derivatives of the given function. 1. f(x,y)= 2e*Y cos 4x + y…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 4xy – 9xyz + 8yz² fx(x,…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 3x²y – 2xyz + 9yz2…

A: The solution is given below in the next step:

Q: Which of the following is not a homogeneous function? h(x,y)= xª − 3xy³ + y¹ − x² cos Option 1 b(x,…

A: Let's solve.

Q: Find the first partial derivative of fx , fy and fz of f ( x , y, z ) = 2xyz +  tan x  +  y sin z…

A: Given problem:- Find the first partial derivative of fx , fy and fz of f ( x , y, z ) = 2xyz + tan x…

Q: find the first partial derivatives ofS. S(x, y) 2x4y - xy + 3y + 1 sr. s) S(x. y) - xe + y sin x ft,…

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

Q: 3 19- Change the order of the integration of S f(1,y)drdy. ŏ y-3 0 -3+z a) S S f(x,y)dydx + § S…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 3x2y – 4xyz + 6yz2 %3D…

A: We have to find

Q: If x, y, and z are lengths of the edges of a rectangularbox, then the common length of the box’s…

A: Given- s=x2+y2+z2 To find- dsdt in terms of dydtand dzdt, if x is constant. Concept Used- As the…

Q: Find a corresponding homogeneous linear DE with constant coefficients whose general solution is Ye…

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Q: i) Let f(x, y, z, p,q) be a homogeneous function of degree n of X, y and Z. Show that the equations…

A: (i) Given that f(x,y,z,p,q) is a homogenous function of degree n of x,y and z. Let We know that to…

Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 4x²y – 8xyz + 7yz?…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 4xy – 6xyz + 10yz f,(x,…

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Q: 3. Give an example of a homogeneous function of the form f(r.y, 2) of degree 3.

A: HOMOGENEOUS FUNCTION A function of the form fx,y,z is said to be homogeneous function of degree 'n'…

Q: 10. Use the chain rule to find the indicated partial derivatives. (a) z = e"Y sin y, x = s+2t, y = ;…

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Q: 1. Find of/ar, af/ay and of/0z (y- sin(x))² f(x, y) = (r - 1) (y+ 2) + S(1, y) = in(V + y²) + (z² +…

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Q: 9- ! VT- Change the order of integration of f(x,y)dxdy. 0 y-1 0 1+x ! Vl- a) f(x, y)dydx +/ /…

A: And we have to change the order of the integration.

Q: 2. Consider the 2nd order linear homogeneous DE x*y" - x (x+2) y' + (x+2)y = 0 for x >0. (a) Show…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 4x²y – 6xyz + 10yz %3D…

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Q: y" + 4y' + 3y = 0, y(0) = 2, y(0) = -1

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Q: Find the all the 1st order partial derivatives of the function f(x, y,z)= Vx² +z-2 + sin

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Q: Solve for the solution of the Homogeneous DE: (x - y)dx + (3x + y)dy = 0 when x = 2; y = -1 a. (x -…

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

Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 2x²y – 5xyz + 10yz?…

A: We will solve the problem

Q: a) a fonction f (xy) is homogeneous of degree 7 in X and y it and only it

A: As per bartleby guidelines we only give first question answers.

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

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Q: 6y" – 5y' + y = 0, y(0) = 4, y'(0) = 0

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Q: (b) Show that the function f: R2 → R given by (* if (x, y) + 0, S(x, y) = 0 if (x, y) = (0, 0), is…

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Q: 1. If z = f(x, y) has continuous second-order partial derivatives and x = r cos 0, y = r sin 0,…

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Q: Let u(x,y,t)=(1/t)exp[-(x²+y²)/4t] where exp denotes e to a function. Show that u(x,y,t) satisfies…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 4x²y – 7xyz + 6yz?…

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Q: 2. i) Let f(x, y, z, p,q) be a homogeneous function of degree n of X, y and Z . Show that the…

A: Given that f(x, y, z, p, q) be a homogeneous function of degree n of x, y and z. To show that z=px +…

Q: a) Find the global maximum and minimum for the function x – xy +y in the square (both inside and the…

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Q: (a) Find the first partial derivative of (i) } f(x, y, z) = (ii) f(x,y, z) = eY² cos xy

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

Q: (a) Find the first partial derivatives of f(x, y) = cos(2.ry) – e³y+2x. (b) Evaluate (0,0) and (T,…

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Q: A function f(x, y) = 2x2 − 5xy is homogeneous of degree n when                      f (tx, ty) = tnf…

A: (a) Consider the given function fx,y=2x2-5xy

Q: y= 0, y(1) = 1, y(1) = 0

A: We have to solve given problems

Q: (x - z)j + (x – y)k tegr S: z = 9 – x² – y², z 2 0 3. F(x, | 4 F(x. y, z) = (-y + z)i + (x – 2)j +…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 4x y - 9xyz + 10yz %D…

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Q: A function f(x, y) = xcos(x + y)/y is homogeneous of degree n when              f (tx, ty) = tnf (x,…

A: fx, y=xcosx+yy a). Now, ftx, ty=tx·costx+tyty=tx·costx+yty=t1·xcosx+yy=t1·fx, y Hence fx, y=xcosx+yy…

Q: 0. Show that if f is homogeneous of degree n, then 1 fz + 2ryfry + y² fyy = n{n – 1)f(x, y). (A…

A: Let f be a homogeneous function of degree n and f has continuous second order partial derivatives.…

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.