Question

Find linearly independent functions that are annihilated by the given differential operator. (give as many functions as possible. use x as the independent variable. enter your answers as a comma-separated list.) d2 − 12d − 45

Answer 1

We want to find all $f(x)$ such that

$(D^2-12D-45)[f(x)]=0$

We have

$(D^2-12D-45)[f]=(D-15)(D+3)[f]=0$

so that either

$(D-15)[f]=0\implies f'-15f=0\implies f(x)=e^{15x}$

or

$(D+3)[f]=0\implies f'+3f=0\implies f(x)=e^{-3x}$

These solutions are linearly independent; we confirm this by computing the Wronskian:

$W=\begin{vmatrix}e^{15x}&e^{-3x}\\15e^{15x}&-3e^{-3x}\end{vmatrix}=-3e^{12x}-15e^{12x}=-18e^{12x}\neq0$

We can also confirm that these solutions are correct by substituting the solutions for $f(x)$ in the differential equation:

$(D^2-12D-45)[e^{15x}]=225e^{15x}-12(15e^{15x})-45e^{15x}=0$

$(D^2-12D-45)[e^{-3x}]=9e^{-3x}-12(-3e^{-3x})-45e^{-3x}=0$