Exercise 6. (Potentials) Calculate and plot the potential function V(r) and iden- tify the equilibrium points and their stability for all the following equations. • i = r(1 – x)

Q: 1) Suppose the charge on a capacitor in a simple electric circuit is governed by the IVP 2y" + y +…

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Q: QI: Solve the equation u = 4 uxr 0<x< 5, Ost<00 that satisfies the conditions u(0, t) = 0, u(5, t) =…

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Q: -mpone (2, 2)

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Q: Find the linear approximation of the f(x, y) = Answer: f(x, y) ~ X x + y at the point (1,6).

A: To find the linear approximation of the function f(x, y) = x/(x + y) at the point (1, 6), we need to…

Q: 4. Given the function f(x, y, z) = x + y x+z find the gradient at P(1,2,3).

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Q: Q10. For the function f(x, y) = x³e3y find the maximum value of its directional derivative at the…

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Q: Consider the system of equations (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal…

A: Introduction: A nullcline is a curve in phase space where a particular variable has zero…

Q: Find the maximum and minimum values of the function f(x, y) = eY subject to r + y = 128 Maximum…

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Q: Find the equilibrium point. D(x) = (x-7), S(x) =x O A. (7,$0) О В. (14,549) C. 2,$12.25 O D. (0,$49)

A: The equilibrium point of two functions D(x) and S(x) is the point which satisfies both equations. In…

Q: Calculate and plot the potential V(x), up to an arbitrary constant, for the ODE d x(t) = =-X.

A: To calculate the potential V(x), we integrate the given ordinary differential equation (ODE) with…

Q: Use the linearization L(3,-2) (x, y) of f(x, y) = -x²y-xy² near the point (3, -2) to approximate the…

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Q: Determine the equation of the quadratic with roots of of -32. +13 -2± √2 and that has a y-intercept

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Q: Let G(x) = ²* w(t) - [² d dt. t4 a Find a function, w, and a number, a, such that G(x) = 4x¯ w(x) =…

A: Given G(x)=∫axw(t)t4dt and G(x)=4x-2-1

Q: Let x) = (x +2) and h 0.3. Find (a) the approximate value of f (2) (b) the true value of f (2) (e)…

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Q: s lineax diggerential Solve nonhomogene ous equation using propesty of diggerential operator I…

A: Auxiliary equation of given ODE is  m3-2m-1=0m2-m-1m+1=0 So we have Complementary solution is…

Q: The function f(x, y) = L(x, y) = arctan(x + 2y) can be approximated by the linear function = ax + by…

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Q: dx Graph the slope field for the equation dt -r(x² - 4) and attach it to the solution. Draw (by hand…

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Q: 12. (D²-3D+2) y = e²x/(1+e²x). 12 (D2 201

A: 12) Given differential equation is  (D2 - 3D + 2)y = e2x/(1+e2x), we solve that problem by method of…

Q: DIfferentiate a) f(x)= cY X-4

A: a) fx=cx-43=cx-413 Differentiate with respect to x using the chain rule f'x=13cx-413-1…

Q: Find all equilibrium points and determine their types when C = 1. * = x, *, = -bx, +1+c(-1+ sat…

A: Given: x.1=x2x.2=-bx2+1+c-1+sat0.5-x1-x2

Q: A particle is moving with acceleration a(t) = 6t + 18. its position at time t = 0 is s(0) 13 and its…

A: Given, at=6t+8 with s0=13 and v0=2. We have to find the position at time t = 14.