1. i) Find partial differential cquation of the family of unit sphere whose centers lic on the line 3x= 4y = 2z. ii) Let f be an arbitrary function and z = := z(x, y). Find the partial differential equation of the family of the surface (z² – x+ y)² = f(x² – xy+ y²).

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: Consider the hyperbolic partial derivative equation Uzz = Uu – sin (x), rE [0, 1], t>0, with the…

A: Given,uxx=utt-sin x , x∈0,1, t>0Boundary conditions: u(0,t)=ux(1,t)=0 , t>0…

Q: Let y1 and y2 be solutions of a second order homogeneous linear differential equation y'+ p(x) y' +…

A: Step-1LetYIandY2(0be solutions of a second orderhomogeneous linear differential equationY't p (x) y'…

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: 4.1 In aerodynamics and fluid mechanics, the functions u(x, y) and v(x, y) in f(2) where f(z) is…

A: We will find out the required values.

Q: 9. Find all the second partial derivatives of Z; given: a) Z = xy³In(3xe ) b) Z = Arctan ( x² /y³)

A: Let us consider the two variable z=f(x,y) The first order partials derivative  is obtained as…

Q: 2) Determine as derivadas parciais de 1ª ordem das funções z=2e² sen(x) P(1,5). (valor: 1,0) no…

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Q: 1 1 f(x,y) = x² - 2xy + 3x +y? %3D 1. Compute the first- and second-order partial derivatives. 2.…

A: Solution is given below:

Q: 11) Find all first-order partial derivatives: 4wz3y\(0 sin w0.25y-3\ f(w, x, y, z) = | 42x5y°42 –…

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Q: (1.-13) Show that the line integral (2xy+3)dx +(x² - 4z) dy-4 ydz is independent of path and (0.0,0)…

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Q: a) z=x² In (x y²) b) z=4 cos(x-3y)

A: As we know that; The Power law: ddxxn=nxn-1 The Product rule: f·g'=f'·g+f·g' The Chain rule:…

Q: 12.2.7. Suppose that Q := {(x, y) e R : x > 0 and y > 0} and that f is a continuous function on R?…

A: Sol

Q: 7. ( ) The surface area of the surface S with equation z = g(x, y), where (x, y) lies in D and f has…

A:

Q: 1. If a function f: DR is differentiable at (a, b)ED, then f has partial derivatives fa and y at (a,…

A:

Q: Answers for 10-12

A: We will solve Q-10

Q: Let y1 and y2 be solutions of a second order homogeneous linear differential equation  y''+ p(x) y'…

A: p(X)= -1 q(X) = -2 Y(X)= c1e2x+c2e-x

Q: The three-dimensional Laplace equation dx = 0 is satisfied by steady-state temperature dy dz…

A: Given that  f(x,y,z)=(x^2+y^2+z^2)^(-1/6) To find the second order partial derivatives

Q: 8 Find all the first partial derivatives of z = 5x°y - 8x y. (Choose all applicable answers.) O zy =…

A:

Q: 4.5.1. Compute the best quadratic approximation to f(r,y) = e" cos y at the point (0,0), f(h, k) z…

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Q: 1. Let z = f(x, y) be defined by the given equation. Find the total differential of z at any point…

A: The solution of the problem is given in the next step.

Q: 28. Find all the second partial derivatives of f(x,y) = x²y + x /ỹ. 29. Find the equation of the…

A:

Q: (a) In electrostatics, the electric field lines can not form any closed loops. However, in the…

A:

Q: 5. Use the separation of variables method to determine the solution u(x; t) of the heat transfer…

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Q: (a) Show that a function u(x,y)=e^(x^2-y^2 ) sin⁡(2xy) satisfies two dimensional Laplace equation…

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Q: Except for the case of constant solutions, it is usually not possible to find explicit expression…

A:

Q: Fiven the following non-homogeneous system of first-order linear differential quation: у3 2у1 + уг +…

A:

Q: 2. (a) Show that F = (e² cos(y), –E sin(y), 2) is conservative using partial derivatives. (b) Find…

A:

Q: (2) Conservatively discretize in space the 2D nonlinear diffusion equation Əu = V• (D(u)Vu) =…

A: Given that, ∂tu=∇Du∇u=∂xDu∂xu+∂yDu∂yu

Q: 9. Establish the fact, widely used in hydrodynamics, that if f(x, y, z) = 0, then A)), (E) - dz -1.…

A: Consider the provided question,

Q: B- Solve the Laplace equation for steady heat conduction in the semi-infinite region 0 < y< 1, x 2…

A: The given problem is to solve the given Laplace equation for steady heat conduction in the given…

Q: ) Solve the partial differential equation problem U = 16uxx, 0 0 Boundary conditions: u(0, t) =…

A: Given PDE is utt=16uxx,   0&lt;x&lt;1,   t&gt;0 Boundary conditions: u(0,t)=u(1,t)=0 Initial…

Q: . Verify that the differential operator defined by…

A:

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: 9. Find all the second partial derivatives of Z; given: a) Z = xy In(3xe) b) Z = Arctan (x² /y®) %3D

A: Given functions for Z are: (a) Z=xy3ln3xey (b) Z=Arctanx2y3=tan-1x2y3. We have to find second order…

Q: 2y" +y – 4y = 0, y(0) = 0, y(0) =1

A:

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

A:

Q: 4. An imaginary ant is walking on an imaginary cartesian plane so that at any point (x, y) it moves…

A:

Q: (d) The partial differential equation a2u a2u = 16 – x2 – 2y for 0 <x < 4, 0 < y< 2 (1.1) ду? is…

A: Given the partial difference equation, uxx+uyy=16-x2-2y, for 0&lt;x&lt;4, 0&lt;y&lt;2. Let  h=k=1…

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: 8. [9 points] By the linear (tangent plane) approximation for the function f(x, y) = Jx² + y3 at…

A:

Q: 7 -4 Consider the surface x'z* + sin(y zª) + 9 = 0. Use implicit differentiation to find the…

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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: 6. Consider the 2-dimensional parameter 41-µ2 plane for the differential equation, * + µi ¢ + µz x +…

A: Use the properties of integration in acceleration, velocity and distance. Use their formulas to…

Q: a) Describe the curve obtained when we make y= 2 and 2= √2 b) Represent on this curve the partial…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Find all the first partial derivatives of z = a sin 3y – 2y? cos 3x (Choose all applicable answers)…

A: Using partial derivatives.

Q: Let f(x, y, z) = xy + z° y = 2rs z = p3 af of ds ' dr Use differentials (not the Chain Rule) to…

A: Given f(x,y,z)=xy+z8put x=s8…

Q: 1-6 Use an appropriate form of the chain rule to find dz/dt. 1. z = 3x²y*; x = r^, y = r² 2. z =…

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Q: The two-dimensional Laplace equation = 0 describes the potentials and steady-state temperature +…

A: given,

Q: 1. For u(x, t) defined on the infinite domain of -o 0, solve ди with boundary conditions: u(x, t)…

A: Given ∂u∂t=-∂4u∂x4+e-x2-t Boundary conditions: u(x,t) and its first and second derivatives in…

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.