2(a) Solve the following differential equations, subject to initial conditions if they are given. Assume y is a function of t. (1) y" + 10y' + 25y = 0 (ii] y" = -64y, y(0) = 30, y'(0) = 6. y" + 8y' - 20y = 0, (iv) y" + 6y' + 13y = 0 2(b] Linear, first order DE s may be solved by the integrating factor method: Write the DE in the form y'+ p(x)y=q(x), thatis, y'+ py=q. Find the integrating factor, I = el p(xjax Then find y from Ty = \\ Iq dx, or by multiplying both sides of y' + py = q by the integrating factor. Solve the DE 2- + By = 5. 2(c) (1) In the identity (2x-1)(x+2) A 2x-1 5, find the values of A and B. (ii) Under certain conditions, the time t (in minutes) required to form x grams of a particular chemical substance in a chemical reaction can be modelled by = = k(1 - x) (2 - x) - 3kx' where x = 0 grams when t = 0 minutes. Show that the differential equation can be written in the form dx ( 2x - 1 ) (x + 2) = - kdt and then solve the initial value problem to find a relation between z and x. Your final answer should be in the form of = f(t), where f(t) is a function of t