Hello. Please answer the following question, ensure that your answer is correct and supported by calculations, explanations and algebraic work. Only utilize the sections/topics seen below. Solutions for Ordinary Differential Equations must be given in real values form(i.e. no complex or imaginary values). Please organize your work, in a reasonably neat and coherent way. Do not use ChatGPT, I will not be able to rate you highly if this is not your work. Thank you so much, I hope you have a great day! Everything necessary for the problem is seen below, it is not incomplete.

Topics: first order linear (autonomous) systems linear systems II (like phase planes and direction fields) linear systems III(like eigenvalues and eigenvectors) linear systems IV (like rewriting undamped oscillators as a system, using eyelets principle and linearity principle, classification of equilibrium point Linear systems V (eigenvalues, trace of a matrix) Systems (evecs)

For the given equation: a" + 3x' + 5x = 0 a) Rewrite in the form: a" = F(ac, ac' ), clearly identifying the expression for F(ac, a'). b) Compute the tangent vectors T'(ac, a' ) at the points (1,-1), (1,0), and (0,-1). c) Which of the following direction fields corresponds to the given equation? X' x' (i) (ii) 2.0 2.0 - 15 1.0 0.5 0.5 X X 0.0 0.0 -0.5 -1.0- -1.0 -1.5 - -1.5 -2.0 -2.0 0.5 1.0 1.5 2.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -2.0 -1.5 -1.0 -0.5 0.0 (iii) X' (iv) X' 2.0 2.0 1.5 10 1.0 0.5 0.5 X X 0.0- 0.0 -0.5 -0.5 - -1.0 -1.0- -1.5 -1.5 -2.0 -2.0 15 2.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 10 15 2.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0