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Chapter 6: Q15E (page 337)

find a differential operator that annihilates the given function.

e2x-6ex

Short Answer

Expert verified

D2-3D+2is the differential operator that annihilates the given function.

Step by step solution

01

Differentiate the function continuously

h(x)=-6exLet the function bef(x)=e2x-6ex

Letg(x)=e2x

Then

g'(x)=2e2x=2g(x)鈬�g'(x)-2g(x)=0鈬�(D-2)[g]=0

Leth(x)=-6ex

Then

h'(x)=-6ex=h(x)鈬�h'(x)-h(x)=0鈬�(D-1)[h]=0

Then

(D-2)[f](x)=(D-2)[g+h](x)=(D-2)[g](x)+(D-2)[h](x)=(D-2)[h](x)=6ex=-h(x)

Then

(D-1)[(D-2)[f]](x)=(D-1)[-h](x)=-(D-1)[h](x)=0

Hence(D-1)(D-2)[f]=0

ThenD2-3D+2is the differential operator that annihilates the given function.

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