Question

Please help answer the 5 parts of this 1 question.

Question 6 -2a is a solution to the following ODE:/" -2/-8y 0. Use Reduction of Order to find a y1 2nd linearly independent solution. [Select] Step 1: Let y- Select] [Se ue(-2x) Then y e-2x) Step 2: Substitu ue-8x) simplify to get [Select e-8x) Step 3: Reduce Step 4: Solve the equation for w. (Select] Step 5: Solve for u. Step 6. Identify the two linearly independent solutions e ae was given as one solution. A second linearly independent solution is tSelect] 1

Answer 1

Y'' - 2 y' - xy = 0 Given that y_1 = e^negative 2x is a solution of the Let, y = v y_1 = v e^negative 2x y' = negative 2 v e^negative 2 x + v' e^negative 2x = (v' -2 v) e^negative 2x = negative 4 v' e^negative 2x + 4 v e^negative 2x + v'' e^negative 2 x therefore y'' - v y' -5y = 0 e^negative 2x [(v'' - 4 v' + 4) - 2 (v' -2v) - 8v] = 0 e^negative 2x (v'' - 6 v') = 0 therefore v'' - 6 v' = 0 (because e^negative 2 x notequalto 0) Let v' = w therefore w' = 6w = 0 therefore log w = 6x + log c v' = c e^6x v = c /6 e^6x + d y = c /6 e^4x + d e^negative 2x 4 e^negative 2 x (20 (u' -2u) e^negative 2 x (3) u'' - 6 u' = 0