Question

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

Answer 1

Answer:

The general solution of the given differential equation

$y = ( c_{1} + c_{2} x ) cos3 x + ( c_{3} +c_{4} x) sin3 x$

Step-by-step explanation:

Step(I):-

Given differential equation

                                   y⁴+18y"+81y=0

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

The auxiliary equation

                               $m^4+18m^2+81 =0$

                              $(m^2)^{2} + 2 (9) m^{2} +(9)^2 = 0$

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

                  ⇒    ( m² + 9 ) ² = 0

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

                      $m^{2} =-9\\m= - 3i and m=3i$

            m² + 9 = 0      

      $m² = -9\\m= -3i and m=3i$

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

Step(ii):-

The complementary function  

              $y = e^{\alpha x } ( c_{1} + c_{2} x ) cos\beta x + ( c_{3} +c_{4} x) sin\beta x$

The general solution of the given differential equation

          $y = e^{0 x } ( c_{1} + c_{2} x ) cos3 x + ( c_{3} +c_{4} x) sin3 x$

The general solution of the given differential equation

   $y = ( c_{1} + c_{2} x ) cos3 x + ( c_{3} +c_{4} x) sin3 x$