Consider crho ∂u ∂t = ∂ ∂x k0 ∂u ∂x ! + αu, where c, rho, k0 and α are functions of x, subject to u(0, t) = 0, u(l, t) = 0, u(x,0) = f (x). assume that appropriate eigenfunctions are known according to the results included in the box on p.157. (a) show that the eigenvalues are positive if α < 0. (b) solve the initial-boundary value problem. (c) discuss lim t→[infinity] u(x, t).

Find the center and the radius of the circle with the equation: x^2-2x+y^2+4y+1=0?

Atriangular field has sides of lengths 21, 39, and 41 yd. find the largest angle. (round your answer to the nearest degree.)

Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions: y < 4x − 8 y is greater than or equal to negative 5 over 2 times x plus 5 part a: describe the graph of the system, including shading and the types of lines graphed. provide a description of the solution area. (6 points) part b: is the point (5, −8) included in the solution area for the system? justify your answer mathematically. (4 points)