(3 points) In this question, you will find a Taylor polynomial approximation of the solution to the differe .al equation: dy _ -2y dx Suppose that we are looking for a solution f(x) = > a;x' to the differential equation that passes through the point (0, 5). 1=1 (a) What is do ? The differential equation requires the function y and its derivative dy Also, two polynomials are equal if and only if they have the same coefficients on like power terms. (b) Use this fact to find a relationship between a, and an . We have a; - (c) Now write a2 in terms of a1 . Then write a2 in terms of ao . We have a2 - do (d) Write a3 in terms of a2 . Then write a3 in terms of ao . We have a3 - a (e) Write a4 in terms of a3 . Then write a4 in terms of an . We have a4 - (f) You should see a pattern: an - (g) Combine your answers from (a)-(e) to write a polynomial approximation of degree 4 of the solution of the the differential equation. (h) Use separation of variables to find the solution of the differential equation that passes throught the point (0, 5). y = (i) Find a Taylor polynomial of degree 4 centred around a = 0 of the solution that you found in (h). P4(X) = GIF