DIFFERENTIAL EQUATIONS: THE BASICS 5 minute review. Remind students what a differential equation is, the difference between ordinary and partial, linear and non-linear, and what the order of a differential equation is. Class warm-up. Differentiate the following functions of x, and construct a linear first-order ordinary differential equation containing a y-term whose solution is the given function. (a) y = x100 , (b) y = x ln x, x (c) y = ee , (d) y = x sin x cos x. Problems. Choose from the below. 1. Differential equations for sin and cos. (a) Show that, for constants a and b, the function y = a cos x + b sin x is a solution to the differential equation d2 y + y = 0. dx2 (1) (b) Find values for a and b in order to give a solution to Equation (1) satisdy = 4 when x = 0. fying y = 3 and dx (c) For which a and b are those solutions to Equation (1) also solutions to 2 dy the differential equation dx + y 2 = 1? 2. Powers. (a) Avoiding backwards reasoning, show that, for constants a and n, the function y = axn is a solution to the differential equation 2 d2 y dy dy + xy 2 = x . (2) y dx dx dx (b) Hence find a solution to Equation (2) with y = 5 at x = 1 and y = 80 at x = 2. 3. Exponentials. (a) Find two values a1 and a2 of a such eax is a solution to the differential equation d2 y dy −7 + 10y = 0. (3) dx2 dx (b) For those values a1 and a2 , show that rea1 x + sea2 x is a solution to Equation (3) for any constants r and s. (c) Hence find a solution to Equation (3) satisfying y = 4 and x = 0. dy dx = 11 at (d) Can you generalise this approach to solve any equation which is of the d2 y dy form p dx 2 + q dx + ry = 0 (with p 6= 0)? Does it split into cases? 1 2 DIFFERENTIAL EQUATIONS: THE BASICS dy dy y For the warm-up, (a) is a solution to dx = 100 x y, (b) to dx = x + 1, (c) to dy and (d) to dx = y x1 + cot x − tan x . There are other options. dy dx = yex Selected answers and hints. 1. In part (c) you should find the condition that a2 + b2 = 1. 2. You should end up with the solution y = 5x4 . 3. You should get a1 and a2 equal to 2 and 5 (in some order), and then the solution in part (c) is 3e2x + e5x . For part (d), we will cover the theory later in the course. For more details, start a thread on the discussion board.
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