3. [10 marks] The second order ODE with time-changing (non-constant) coefficients of the form dt2 + 2 d -7 + At y + By = 0, t >0 (* ) where A and B are real constants, is called the Euler equation or Cauchy-Euler equation. In this problem, you will solve this type of equations. (a) Introduce the new variable x = Int (t = e) and calculate d and day i atz in terms of dy and dry by using the chain rule), then show that equation (*) can be transformed to an equation with constant coefficients of the form day + dx2 + (A - 1) dx dy + By = 0 (b) Use part (a) to find the general solution of the following equation. dt2 dt + 3t - - 3y = 0 (Make sure that in the end your solution is a function of t and not x.) (c) Use part (a) to transform the following non-homogeneous Euler equation to a non-homogeneous equation in x and then find a particular solution for the non-homogeneous ODE. dt2 + 3t it atdy - 3y = t2 In(t) (Hint: The non-homogeneity term must be transformed to a function of x as well.) (d) Use part (b) and (c) to find the general solution to the equation dt2 dt + 3t - 3y = t2 In(t) with initial conditions y(1) = 1, y'(1) = - 1 25