Suppose we have a bar of length L = 2 and, instead of keeping the ends at a constant 0°C, the bar has insulated ends. We can model this situation with the following boundary conditions: Uz(0, t) = 0 , uz(2, t) = 0 (a) Show that the boundary condition uz (0, t) implies X'(0) = 0 or T(t) = 0. (b) Fill in the blanks: The boundary condition u (2, t) = 0 implies = 0 or = 0. (c) This leads us to the boundary value problem with param ter: X" + XX = 0, X'(0) = ), X'(2) = 0 Find the nontrivial solutions to the BVP for the case w rere A > 0. Assume A = µ?. Recall that sin 0 = 0 for 0 = nr.

Q: 2. Solve the boundary-value problem = 4, y(0,t) = y(5,t) = 0, y(x,0) = 0, y.(x,0) = 5 sin rx. %3D

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Q: 2. a). Find the velocity and acceleration of a particle whose position function is x(t)=…

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Q: 3. a.) Find the partial derivative of the function f(x, y) = x² tan y + In( sin xy) 1 b.) IF ƒ(x, y)…

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Q: 1. The temperature at the point (r, y) on an unevenly heated plate is given by T = f(r, y) = 100+…

A: Given f=100+5x2+15x+10y2gradient of f∇f=∂f∂xi+∂f∂yj     =10x+15i+20yjAt (2, 1)∇f=10x+15i+20yj…

Q: 4. Let f(x, y) = 2TY x = In 2 g and y = In s. . a. Find the equation of the tangent plane to f(x, y)…

A: Let f(x,y)= 2xy , x=t/ln2 and y= lns Here we have to find the equation of the tangent plane to…

Q: S[y] = [ "_ dx (y^² + y²), _y(0) = 1, y(v) = v, v > 0, is given by y = coshæ + B sinh x, 0 < * Συ,…

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Q: d) A particle of mass m in a rectangular box with a fixed volume has ground state energy k2 E(x, y)…

A: A function has local minimum at a point if the gradient of function is 0 at that point and hessian…

Q: Q.1. Justify why the level curves very close to the origin (0, 0) for the function f(r, y) =…

A: To find the level curves, equate the function fx,y with a constant C.

Q: 2. Find the lengths of the given curves: a) y = x/2 -x2,1sK< 4 L = V1+f(x)*dx 3

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Q: Find a parametrization of the tangent line to r(t) = (In(t))i + t-°j+ 3tk at the point t = 1. (Use…

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Q: Consider the surface F(x, y, z) = x²28 + sin(y¹28) — 4 = 0. Find the following partial derivatives =

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Q: circle. -) Use the Lagrange multipliers to find the local extrema of the function f(x, y) = 4x² + y²…

A: 4(b) Given function is fx,y=4x2+y2+z2 Subject to the constraints 2x−y+z=4 and x+2y−z=1 We have to…

Q: 17. y = e"(p + p\p)

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Q: For the function f (x,y) = 6y² +6xy-2y +3x Compute the discriminant D= Use the 2nd Derivative Test…

A: Given : f(x, y) = 6y² + 6xy - 2y³ + 3x²

Q: Dse Runge - kutta method to Solve the equation: dly x +y %3D dx at x =0.8 given that using h= o.4 y…

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Q: Since y"|? v0 at x = -1/2 e we have a relative minimum at the point (c, f(c)). Substitute c = 5e¬1/2…

A: Solve it

Q: Draw the level curves of f(x,y)= x² + 9y² for k=0, k=1, k=9.

A: Given curve is, fx,y=x2+9y2

Q: Solve the following IVP using the substitution v = x + y : dy + 1 = ex+y, y(1) = -1. dx

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Q: In (21y) = 7x + 21y is an equation that defines y as an implicit function of x, one of th…

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Q: r(t) = (x(t), y(t), z(t)) where x(t) = - 2t + 3t + 3 y(t) = 2t – 2 z(t) = 2t + 5t

A: Step:- 1 Curvature κ=T'(t)r'(t); T is tangent vector, k=r'(t)×r''(t)r'(t)3 Given that, r(t)=x(t),…

Q: b. Solve by using variable separation method. = 0 ax2' ay2 Which satisfies the conditions, u(0, y) =…

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Q: Let f(x, y) = √x² + y² - 4 y 2 - Identify the equations of the level curves of f for k = 0, 1.…

A: A scalar function f(x,y) is a rule that is used to associate a number or quantity with a particular…

Q: (ztan 2 +1) dx – x dy = 0

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Q: Given z = f(x, y) = Va2 + y² + 4y. %3D (a) Find its domain and range. (b) Sketch the level curves…

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Q: d²y – 4y = 0 dy, - (at x = 0) = 4 dx а. dx? y(0) = 2; - d²y b. 5- dx2 sdy dy - (at x = 0) = 0 dx +…

A: To solve the given differential equations.

Q: h(x, y) = xy2/3 + xy x⁹ +2y³ 1 √3' us at the point (0, 0) or not. If not, determ

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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.