4. [2b] Find the steady state distribution of temperature in the cylinder u[r,1t]}oo ' _ . z=1 {Hr-:2 l | '2 u[2.z]= | l I \"Udall[ll E 2:0 Derel l l l | r20 r: given that the sides and bottom are kept at zero temperature and the top remains at a constant of temperature [In using the following diffusion equation in cylindrical coordinates: 2 2 3_u+ll_u+g= lb2 1' 31' 322 a. Use the technique of separation of variables wit u[t'.E]= R[r]E[z] to get equations for R and Z. Let the separation constant be \"1:2 \". b. Solve the differential equations in R and E. c. Apply the boundary conditions above (in red). Also [as a forth condition} note that the temperature remains bounded at r = fl. [Hint I: In applying conditions for the function thJ at some point you need to rewrite your solution in terms of a \"real"~ hyperbolic function this substitution should be obvious after looking at the definitions of these I'Llnetions.] [Hint 2: Finally your superposition solution will look like a series in Bessel functions {see Section 9.5 Eqs. 9.3? and 9.33} for which you need to solve for the coefficients. f'it this point you do not need to solve the integral for the eoetlieients.just set it up!] [Extra eredit lfl] Solve the integral equation for the series coefficients from part c. hint: You will need to use the following Bessel function identity 3 Eli'll'lllll'l