(Solved): (2 marks) Consider the Maclaurin series fore and cosh z: where A = || e2x - C= IM8 (i) Using the p ...

(2 marks) Consider the Maclaurin series fore and cosh z: where A = || e2x - C= IM8 (i) Using the power series above, it follows that the Maclaurin series for e2x by and cosh z = = A + Br + C?² X P3(x) = B z2k (2k)! + Dz³ + || and D= is given (ii) Using the power series above, or otherwise, calculate the Taylor polynomial of degree 3 about 0 for e² coshz. [Make sure to use Maple syntax when you enter the polynomial. For example, for P3(x) = 4+3x+5x² + 7x:3 you would enter 4+3*x+5*x^2+7*x^3.] (3 marks) The general solution of the homogeneous differential equation d² 24 (2) ? 8 —y (2) + 16 y (z) = 0 da is given by where a = 4 and A and B are arbitrary constants. O (ax + b)e5z aze5x 5x ae 4 az²e5z uh a) What form would you take as your guess? = Aeªz Let us now find a particular solution to the non-homogeneous differential equation d² dz2(z)-8- y (z)+16y (z) = 4e5r? up + Bre ba and b = b) Enter your particular solution up (of the above form, evaluating a and/or b) in the box below.