Question

Solve the equations by the use of the operator D

D²y -6Dy +9y= e^3x + e^-3x

Answer 1

$D^2y-6Dy+9y=e^{3x}+e^{-3x}$

For yn

$(D^2-6D+9)y=0\\m^2-6m+9=0$

$\implies(m-3)^2=0\\\implies m=3,3$

$\\yn=\frac{1}{D^2-6D+9}(e^{3x}+e^{-3x})$

$\\=\frac{e^{3x}}{D^2-6D+9}+\frac{e^{-3x}}{D^2-6D+9}$

$=\frac{xe^{3x}}{2D-6}+\frac{e^{-3x}}{36}$

So solution is $y(x)=y_n+y_p$

$\implies y(x)=ae^{3x}+c_2xe^{3x}+\frac{x^2e^{3x}}{2}+\frac{e^{-3x}}{36}$