Question

In Problems 27–34 find linearly independent functions that are annihilated by the given differential operator.
${\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{6}\mathbf{D}\mathbf{+}\mathbf{10}$

Short answer

$e^{3x}\cos x\text{and}{e}^{3x}\sin x$

Step-by-step solution

Definition of linearly independent functions

Two functions y₁ and y₂ are said to be linearly independent if neither function is a constant multiple of the other.

Evaluation

Now,$D^2-6D+10=D^2-2\left(3\right)D+\left(3^2+1^2\right)$

From (7) of section 4.5,

$D^2-2\left(3\right)D+\left(3^2+1^2\right)\{\underset{e^{3x}\sin x}{e^{3x}\cos x}=0$

Thus,$e^{3x}\cos x\text{and}{e}^{3x}\sin x$are required functions that are annihilated by $D^2-6D+10$.

Thus, role="math" localid="1668170852869" ${\mathit{e}}^{\mathbf{3}\mathbf{x}}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathbf{}\mathbf{\text{and}}\mathbf{}{\mathit{e}}^{\mathbf{3}\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}$are required functions.