Problem 1 In this problem we use the method of reduction of order to study Euler's equation x y"(x) + ary' (x) + By(x) = 0. (0.1) Plugging the general form y(x) = x" in (0.1) leads to x"(r(r - 1) + ar + B) = 0, which implies that r must be a root of the indicial equation; that is, a solution of F(r) := r(r - 1) +ar+ B=0. (0.2) a) Assume that r1 = 12 are roots of the indicial equation (0.2). We know that y1 () = x"l solves Euler's equation (0.1). To find another solution y2(x), assume it can be written as y2(x) = v(x)yl(x) = v(x)x71. (0.3) Plug y2(x) given in (0.3) in Euler's equation (0.1), and write down a differential equation, where only v'(x) and v"(x) appear. b) Use the change of variable u(x) = v'(x) and write down a first order separable equation with u(x) as the unknown. Solve for u(x). Hint: Use the fact that F(r) = r(r -1) +ar + B = (r -ri) since ri is a repeated root of F(r) = 0. c) Having found what u(x) is and using that u(x) = v'(x), solve for v(x). Use this v(a) to obtain a second solution as in (0.3). Write down the general solution y(x) of (0.1) as a linear combination of the two solutions found above. d) Find the general solution to x y"(x) - 3xy' (x) + 4y(x) = 0