Find the general solution of the given differential equation. y^(4)+18y"+81y=0

The general solution of the given differential equation

Step-by-step explanation:

Step(I):-

Given differential equation

                                   y⁴+18y"+81y=0

                         ⇒    (D⁴+18D²+81)y =0

The auxiliary equation

        we will use formula  ( a + b)² = a² + 2 a b + b²    

                  ⇒    ( m² + 9 ) ² = 0

                   ⇒    ( m² + 9 ) ( m² + 9 ) = 0

            m² + 9 = 0      

The complex roots are   0± 3 i ,0 ± 3 i

Step(ii):-

The complementary function  

The general solution of the given differential equation

The general solution of the given differential equation