Solve the given differential equation by undetermined coeffi

Solve the given differential equation by undetermined coefficients y\'\'-8y\'+16y=24x+2

Solution

Given that

y\'\' - 8y\' + 16y = 24x + 2

d²y/dx² - 8dy/dx + 16y = 24x + 2................................1

The D-operator form is ,

( D² - 8D + 16 ) y = 24x + 2

The auxialary equation is,

   m² - 8m + 16 = 0

    m² - 4m - 4m + 16 = 0

   m ( m - 4 ) - 4 ( m - 4 ) = 0

         ( m - 4 ) ( m - 4 ) = 0

         m = 4 , 4

       m₁ = 4 , m₂ = 4

If the roots are real and equal then the complementary function is ,

                                                                      y_c = c₁ e^m₁x + c₂xe^m₂x

                                                                                                     y_c = c₁e^4x + c₂xe^4x

For a non homogeneous term 24x + 2 assume perticular solution is,

                                                         y_p = Ax + B

Substitute y = Ax + B in equation 1

                d²y/dx² - 8dy/dx + 16y  = 24x + 2

              d²/dx² ( Ax + B ) - 8d/dx( Ax + B ) + 16 ( Ax + B ) = 24x + 2..............................2

d/dx( Ax + B ) = A.d/dx(x) + d/dx( B ) = A + 0 = A                                       [ since,d/dx( x ) = 1]

d²/dx²( Ax + B ) = d/dx( A) = 0                    [ since , d/dx( constant ) = 0 ]

From equation 2

            d²/dx² ( Ax + B ) - 8d/dx( Ax + B ) + 16 ( Ax + B ) = 24x + 2

                         0 - 8A + 16Ax + 16B = 24x + 2

                                16Ax + ( -8A + 16B ) = 24x + 2

           Equating the coefficients

                    16A = 24

                        A = 24 / 16

                        A = 3 / 2

-8A + 16B    = 2     

-8(3/2) + 16B = 2

   -12 + 16B = 2

             16B = 2 + 12

             16B = 14

                 B = 14 / 16

                  B = 7 / 8

Hence,

Perticular solution is ,    

                     y_p = Ax + B

                    y_p = ( 3/2 )x + ( 7/8 )  

The general solution is , y(x) = y_c + y_p

                                        y(x) = c₁e^4x + c₂xe^4x + ( 3/2 )x + ( 7/8 )  

Therefore,

    The general solution is ,   y(x) = c₁e^4x + c₂xe^4x + ( 3/2 )x + ( 7/8 )