The temperature distribution u(x, t) in a bar 1 m long, insulated along its length, is given by the heat equation, Ju J²u Əx² Ət (0 < x < 1, t > 0), where x measures distance along the bar, t is time, and u is temperature measured in °C. Initially the bar has the temperature distribution (0 < x < 1). The end of the bar at x = 0 is kept in contact with a block of ice at 0° C, while the other end at x = 1 is insulated. u(x, 0) = cos² (577) 2 sin ( 5πx 2 (a) Write down the boundary conditions for the temperature distribution that model the situation described above. 5πα 2 (b) Use trigonometric identities to show that the initial condition can be expressed as u(x, 0) = P sin + Q sin (² and find the real numbers P and Q. (c) Consider the eigenvalue problem 15πα X"(x) + XX(x) = 0,_X(0) = X′(1) = 0. Find the eigenvalues and eigenfunctions. You may assume that all eigenvalues are positive. (d) Use the method of separation of variables to find an expression for the temperature distribution in the bar for t > 0.

Q: 6. An ant crawls along a linear path with a velocity of v(t) = cos(3t) inches per minute. (a) What…

A: Given, Velocity of the ant  v(t) = cos(3t) Let's assume the initital position = x = 0  at time t = 0…

Q: Find y' at the point (6, 6) where x sin(x? – y²) + 33 = x? – 3.

A:

Q: Solve for d) and e). The answers for a) b) c) I have posted on barterby previously and received.

A: Step 1: Step 2: Step 3: Step 4:

Q: '] Find the derivative of the function. a) y=c sint+\t cos t b) y = secx+e

A: Tje solution of the problem is given in the next steps .

Q: 8. For the function y = e* + sinx: a. Determine y', y",y",y^),y(5), y(6). b. Describe the pattern…

A: We have to find the differentiations of the give function and find y101.

Q: b.) If r(x)= 1 √g(2x) > find r'(x) at x = 2

A: Topic = Derivative

Q: A hamster wheel has a point on the wheel with height modelled by the equation, where h is in cm and…

A: The instantaneous rate of change of a function is the change in function for a very small change in…

Q: If xy + e' = e, find the value of y" at the point where x = 0.

A:

Q: A particle's acceleration is given by the expression a(t) = = 2t3 m/s². The particle was measured to…

A: