An infinite solid circular cylinder is initially at a uniform temperature of 150° C. At time t = 0 the temperature around the entire boundary is suddenly reduced to 0°C, and maintained thereafter. Determine the temperature at any point of the region at any subsequent time.

Find the value of c such that f(x, y, t) = e^ct.sin(4x) cos(2y) satisfies the heat flow equation ft = fxx + fyy. c=

Decide whether following functions satisfy the wave equation utt =a² uxx.

Given f(x, y, z)= 18 x y z +18, use the total differential to estimate the change in f from the point P = (-4,-8,-4) to (-4.02,-8.02, -3.96). " df =

Expand the function f(x, y) = e sin(y) - 5 at (In(4), 0) by Taylor's formula up to and including the second-order terms. f(x, y) Z

Suppose,y is a function of x, i.e. y = y(x), and -9.xy+3.e+y = -3. a. Find dy/dx in terms of x and y.

1. A horizontally stretched string of length L = 7 is subject to a nonuniform body force per unit mass Q(x, t), so that the resulting vertical displacement u(x, t) satisfies J²u 21² and the initial conditions are J²u əx² (1) where e is a constant. The ends of the string are looped around and free to slide on vertical frictionless wires so that the boundary conditions are + Q(x, t) Ju(0, 1) — 0, Ju(π, l) Әх ər u(x,0) = 0, Ju ƏL = 0. (x,0) = Vo where Vo is a constant. (a) In (1), what are the dimensions of c? What does e represent physically? Next, seek a solution to (1-3) in the form u(x, t) = Σan (1)on(x) n=0 (2) (3) (4) by following the steps below: (b) Assume a separable solution to the homogeneous version of the PDE and boundary con- ditions (1)-(2) of the form u = o(r)h(t). Write down or find the eigenvalues An and eigenfunctions on(r) of the homogeneous problem. (c) Using the eigenfunction expansion (4) and your results from (b), find the second-order ODEs satisfied by each unknown time-dependent amplitude an (1) for Q(x, t) in (1) given by Q(x, t) = cos + et cos(2x) (5) where is a constant. (d) Find the initial conditions for an (l) for all n = 0, 1, 2.... (e) Using the ODEs and initial conditions found in (c) and (d), obtain the general solution for an (1) for all n = 0, 1, 2.... (f) Write the final expression for u(x, t).

2. A support for electrified railway cables is cantilevered from the side of the track by a beam with spring stiffness k. The mass of the beam is 3M and is assumed to be concentrated at the free end. The cable, of mass 2M, is supported by a spring of stiffness k from the end of the cantilever. The system of equations governing the motion of the system is: 3 M y_{1}^{\prime \prime}=-2 k y_{1}+k y_{2} 2 M \ddot{y}_{2}=k y_{1}-k y_{2} k = 22 Write the above system of differential equations in matrix form. Then, by considering the trial solution: y = e"X, show that system can be written as an eigenvalue problem. (3) b) Find the general solution for the system of equations by solving the eigenvalue problem. (12)

4. Find the solution and final value (when t→ 0) of the following differential equation by using Laplace transform. Include the mathematical method to justify your solution and state any properties of Laplace Transforms that you have used to perform your manipulations. (12) \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+c x=2 e^{-4 t} \text { Where } \frac{d x}{d t}=1 \text { and } x=0 \text { when } t=0 b=3 \text { and } c=3

\text { If } f(x, y)=x^{2} y-3 y^{4}, \text { find } f_{x}(1,0) \text { and } f_{y}(1,-1)