Question

In Problems 27–34 find linearly independent functions that are

annihilated by the given differential operator.

${\mathit{D}}^{\mathbf{2}}\mathbf{-}\mathbf{9}\mathit{D}\mathbf{-}\mathbf{36}$

Short answer

The linearly independent functions are $e^{12x}\text{and}{e}^{-3x}$.

Step-by-step solution

Annihilator Operator

IfLis a linear differential operator with constant coefficients and f is a sufficiently differentiable function such that

$L\left(f\right(x\left)\right)=0$

then L is said to be an annihilator of the function.

Using the Differential operators 

The linear differential operator $(D^2-9D-36)$which isis in the form$(D-\alpha )(D+\beta )$, then it can annihilate the independent functions that in the form$e^{-\alpha x}\text{and}{e}^{-\beta x}$, then we can have the functions which are annihilated by the given operator as,

$e^{-(-12)x}\text{and}{e}^{-3x}$

Which is

$e^{12x}\text{and}{e}^{-3x}$.

Therefore, the linearly independent functions are role="math" localid="1668409296116" ${\mathit{e}}^{\mathbf{12}\mathbf{x}}\mathbf{}\mathbf{\text{and}}\mathbf{}{\mathit{e}}^{\mathbf{-}\mathbf{3}\mathbf{x}}$