3. Let p: R → R be a differentiable function. Prove that the equation Uų = p(u)u, t > 0 has a solution satisfying the functional relation u = entiable function. In particular find such solutions for the following equations: f(x+p(u)t), where f is a differ- (a) ut = (b) u = ux; (c) u = u sin(u)uz. (Note: u = = u(x, t), so the chain rule applies to p(u) as well.)

Q: consider a function f, defined by f(x,y) = -y/√x2+y2   what is the behavior of f along the line…

A:

Q: b) Consider the second-order truncation error (0 (Ax)2) with the step size (-Ax) and then, obtain a…

A: The truncation error is O∆x2 and the step size is -∆x

Q: A r&r.'Find y as a function of t if 9y" – 24y' + 7y = 0, and y(5) = 3, y' (5) = 8.

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Q: (b) Verify that the following function is harmonic u(x, y) = −e sin x - 2e-siny. Find a harmonic…

A: Here we have to verify that the function is harmonic and then find the harmonic conjugate.

Q: 2. Find the directional derivative of f(x,y) = x cos y in the direction of v = 22, -1). A. (2 cos y…

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Q: Q10

A: Given F(u,v)be a function of two variables. let Fu(u,v)=G(u,v) and Fv(u,v)=H(u,v) Here f(x)=F(u,v)…

Q: Let f(x, y, z) = 5x³ - y³ + z². Find the maximum value M for the directional derivative at the point…

A: Our objective is given below:

Q: 15.8 Consider the equation x³ + 3y² + 4xz² – 3z²y = 1. Does this equation define z as a function of…

A: Given:  The equation: x3 + 3y2 + 4xz2 - 3z2y = 1. To define: z in the equation as a function of x…

Q: Solve ux – uy +yu = 0, where u is a function of x and y. %3D

A: Given: ux-uy+yu=0. The objective is to solve u, where u is a function of x and y.

Q: Find the gradient Vf and the directional derivative at the point P (1,–1,2) in the direction a =…

A: The given function f(x, y, z) = x3z - yx2 + z2.The gradient of a function f is defined as∇f(x, y, z)…

Q: 3. Consider f(x)=r. For which range of values will the function converge or diverge?

A:

Q: The position of an object at time t is r(t) = (7 + t)i + e'j. Eliminate the parameter 1 to find y as…

A: Given the position of an object at time t is r ( t ) = ( 7 + t ) i + et j . We eliminate the…

Q: In physical chemistry, it is shown that the pressure P of a gas is related to the volume and…

A: Given- In physical chemistry, it is shown that the Pressure P of a gas is related to the volume and…

Q: Verify that the function u, defined by u(t, x) = t sin(x – t²), is a solution to the equation ,2 U +…

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Q: Prob3: Find the derivative of the function at P, in the direction of A f(x,y,z) = (1.0.). = cos xy +…

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Q: n For each nEN, consider x = with x ER+. Define the sequence {gn} by x (a) (b) A = [1,2) CR. (1)…

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Q: c) Consider the third-order truncation error (0 (Ax )³) with the step size Ax and also (-Ax) then,…

A: The truncation error is of order O∆x3 The step length are ∆x, -∆x

Q: Look at this graph: 10 6. 8. 7 5. 0. 4 5 6 7 8 9 10 What is the slope? 2. 4. 3. 2. 1.

A: In this question, concept of slope is used. Slope The slope of a line is a measurement of a…

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Q: ) Find an analytic function f(z) = u(xr, y) + iv(r, y) with real part -. -----) u(r, y) = x? – y? +…

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Q: 1. Consider the function f(r, y) = 70 – 4.x2 – y2. (a) Find the tangent plane to f(x, y) at (-3,…

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: