Use the method of undetermined coefficients to find solution

Use the method of undetermined coefficients to find solutions to the following non-homogeneous, constant coefficient, linear ODEs.

y?? +6y? +9y = te3t

Solution

The complementary solution is yc = C1 e 3t + C2 te 3t . g(t) = e 3t , therefore, the initial choice would be Y = Ae 3t . But wait, that is the same as the first term of yc , so multiply Y by t to get Y = Ate 3t . However, the new Y is now in common with the second term of yc . Multiply it by t again to get Y = At 2 e 3t . That is the final, correct choice of the general form of Y to use. (Exercise: Verify that neither Y = Ae 3t , nor Y = Ate 3t would yield an answer to this problem.)

Once we have established that Y = At 2 e 3t , then Y ? = 2Ate 3t + 3At 2 e 3t , and Y ? = 2Ae 3t + 12Ate 3t + 9At 2 e 3t . Substitute them back into the original equation

(2Ae 3t + 12Ate 3t + 9At 2 e 3t ) ? 6(2Ate 3t + 3At 2 e 3t ) + 9(At 2 e 3t ) = e 3t   

  2Ae 3t + (12 ? 12)Ate 3t + (9 ? 18 + 9)At 2 e 3t = e 3t

  2Ae 3t = e 3t   

  A = 1 / 2

Hence,Y(t)=1/2t e 2 3t

therefore, y=C1 et3+C2 te3t+1/2 t2 e3t