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

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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: 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: (ztan 2 +1) dx – x dy = 0

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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.