Question

2) (25 points) a) (5 points) Verify that y= eat is a solution of the homogeneous differential equation y" - 12y' + 36 y = 0. b) (15 points) Use the method of reduction of order to find a second solution 72 of the given homogeneous equation and a particular solution y of the nonhomogeneous differential equation y" - 12y' + 36 y = 36. e) (5 points) Can you write the general solution of the nonhomogeneous differential equation y" - 12y' + 36 y = 36? Explain.

Answer 1

Using reduction method we find the general solution of the given differential equation.
2. о) Given the differential equation, y" lay't 36 Given that, yea eft - О 1 = 6е вх 4," = 36 ебя Hotо, "- 12' +36 4) а 36 евя. 12. ве", жебх 12e6772e6x t (1

(2) 6x 3 еб." t * 7"- 12 + 6 =0 Yo Satisfies the equatton o " Y = eft is a solution of the homogeneous differential equation. Ya be the another solution. 44. 42 6 = ed- я 6e61, + e“ #d" - 36 ев* + ев", se“.. 28" - 36e61% 12e“, + е**." Given the Norhomogeneous equation is а"- 12+ + 264 = 36 Putting the value of ylsylry ebay , 12 ( себЧ.+ ев) + 36 36 5 эсебя. +1г.в. + е**– зав. - 134. заева. – 136 + зувчу. = 30 ebayan - Я. * 6) || 12 82 + 12e“. + - 28 - 6а = 36e 2". Integrating both side 2' = Jэсебя te [ 9 is an integrating integrating constant . - Зве 67 - 6 26th 2 ' . а 42 - J-sel + Jade [ Integnating both side]

→ Y2 = be takty [ C2 is an arbitrary constant 42 = è 6% 4 6x + 2 froom 261(6-67742+6) => y = Gret07962+1 which is the general solution of the nonhomogeneous equation.