Question

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

${\mathbf{D}}^{\mathbf{3}}\mathbf{-}\mathbf{10}{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{25}\mathbf{D}$

Short answer

$1,e^{5x},x{e}^{5x}$

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

We have

$$\begin{gathered} D^3-10D^2+25D \\ {D}^3-10D^2+25D=D\left(D^2-10D+25\right) \\ =D(D-5{)}^2 \end{gathered}$$

We know that$D\left(1\right)=0$

And differential operator $(D-\alpha {)}^n$annihilates each of the functions

$e^{\alpha x},x{e}^{\alpha x},x^2{e}^{\alpha x},\dots \dots \dots ,x^{n-1}{e}^{\alpha x}$

Then with $\alpha =5,n=2$, we see that

$$\begin{gathered} (D-5{)}^2\left(x{e}^{5x}\right)=0 \\ (D-5{)}^2\left(e^{5x}\right)=0 \end{gathered}$$

Thus, $1,e^{5x},x{e}^{5x}$are annihilated by $D^3-10D^2+25D$.

Thus, role="math" localid="1668170404053" $1,e^{5x},x{e}^{5x}$
are required functions.