Solve the nonhomogeneous equation y 3y 2y 4x2SolutionAuxi

Solve the nonhomogeneous equation y\" - 3y\' + 2y = 4x^2

Solution

Auxillary equation is D²-3D+2=0 ; (D-2)(D-1)=0; D=2 or D=1

Complementry function is C.F.=C₁e^2x+C2e^x

Wher D=d/dx

Now PI=1/F(D)*x²

=1/(D²-3D+2)*x²

   ⁼1/2*(1+(D²-3D)/2)^-1 *x²

        = 1/2*(1-(D²-3D)/2 +((D²-3D)/2)² +....Higher order of D)*x²

=1/2*( 1-D²/2+3D/2 +1/4( D⁴-6D³+9D² )+......Higher order of D)*x²

  =1/2*(1-D²/2+3D/2 +9D²/4+........Higher order of D)*x²

=1/2*(1+3D/2 +7D²/4+........Higher order of D)*x²

      =1/2*( x² +3x + 7/2 )

Total solution = CF +PI =C₁e^2x+C2e^x +1/2*( x² +3x + 7/2 )