Question

In Problems 27–34 find linearly independent functions that are annihilated by the given differential operator.

${\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{5}$

Short answer

$\cos \sqrt{5}x\text{and}\sin \sqrt{5}x$

Step-by-step solution

Definition of linearly independent functions

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

Evaluation

Now,$D^2+5=D^2+(\sqrt{5}{)}^2$

From (7) of section,

$\left[D^2+{\left(\sqrt{5}\right)}^2\right]\left\{\begin{array}{l}\cos \sqrt{5}x\\ \sin \sqrt{5}x\end{array}\right.=0$

Thus, role="math" localid="1668408832112" $\mathit{c}\mathit{o}\mathit{s}\sqrt{\mathbf{5}}\mathit{x}\mathbf{}\mathbf{\text{and}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\sqrt{\mathbf{5}}\mathit{x}$
are required functions.