I am currently working through a chapter in my textbook,

sunyerneq

sunyerneq

Answered question

2022-02-25

I am currently working through a chapter in my textbook, and it is on solving a form of linear differential equation.
I don't understand how ddtek(t)}x=0, since it says that ddtek(t)=p(t)ek(t). Surely this would mean that ddtek(t)x=xp(t)ek(t).

Answer & Explanation

Gene Espinosa

Gene Espinosa

Beginner2022-02-26Added 7 answers

Start from ek(t)dxdt+p(t)ek(t)x=0
Note that p(t)ek(t)x=ddtek(t)
Therefore, by the product rule
ddt(ek(t)x)=ddt(ek(t))x+dxdtek(t)=p(t)ek(t)x+ek(t)dxdt=0
Let me know if this clears up the confusion
e4mot1ic5bf

e4mot1ic5bf

Beginner2022-02-27Added 6 answers

Note that
ddt(ek(t)x)=ek(t)dxdt+ddt(ek(t))x
=ek(t)dxdt+p(t)ek(t)x
according to the product rule. So if we run through the lines in reverse order, we have the derivation.

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