3. Consider ay + by' + cy = f(x) where a, b, c ER, a 0, and y = y(x). (a) Given y(x0) =

3. Consider cry\" + by' + cy = f(I) Where a, b, c E R, :1 34E 0, and y = 11(1). (a) Given y{.:r;g] = Y0 and y'(3;0] = Y1, when does a unique solution exist? {13) Supposing a solution 3; exists, describe the relationship between y",y',y and 3'. Use a theorem [or show that the denition applies] to justify this relationship. (c) Suppose fi(r) = In(x). Construct a linear operator A so that A[fi] = 0. Then find all solutions y of the equation Aly] = 0. Is it possible to find a particular solution to the constant-coefficient ODE using method of annihlators? Justify your answer and build off your work in part (b). (d) Suppose f2(x) = cosh(x). Construct a linear operator A so that Alf2] = 0. Then find all solutions y of the equation Aly] = 0. Is it possible to find a particular solution to the constant-coefficient ODE using method of annihlators? Justify your answer and build off your work in part (b). (e) Solve ONE of the following equations using two different methods. Reconcile your an- swers. Is there a reason why you chose to solve one or the other? y" - 2y' ty = fi(x) OR. y"- 2y ty = f2(x)

3. Consider ay" + by' + cy = f(x) where a, b, c ER, a 0, and y = y(x). (a) Given y(x0) = Yo and y'(x0) = Y, when does a unique solution exist? (b) Supposing a solution y exists, describe the relationship between y", y', y and f. Use a theorem (or show that the definition applies) to justify this relationship.