Question

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

${\mathbf{D}}^{\mathbf{2}}\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{5}\mathbf{)}\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{7}\mathbf{)}$

Short answer

$1,x,e^{5x}\text{and}{e}^{7x}$

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 know that$D^2\left(x\right)=0$

So$D^2\left(1\right)=0$

From (5) of section

With a = 5 and$n=1;(D-5)e^{5x}=0$

With a =7 and$n=1;(D-7)e^{7x}=0$

Thus,$1,x,e^{5x}\text{and}{e}^{7x}$are annihilated by $D^2(D-5)(D-7)$.

Thus, role="math" localid="1668169893696" $1,x,e^{5x}\text{and}{e}^{7x}$
are required functions.