Why doesn't Zwiebach's solution use the equations of motion for problem 8.6?

[ehrenfest]

Homework Statement

Here is Zwiebach's solution to Problem 8.6.

Can someone explain the sentence "we do not use the equations of motion".

I used the equations of motion and I got a completely different answer!

Homework Equations

The Attempt at a Solution

 

[Jimmy Snyder]

Show what you have done.

 

[ehrenfest]

Sorry. I stupidly forgot to upload the attachment. Anyway above the parenthetical that says note that we do not use the equation of motion, I used the equation of motion to get

$$\delta L = \epsilon \frac{d}{dt}(\frac{\partial L}{\partial \dot{q}} \dot{q} )$$

And the rest is totally different for me.

So, why can I not use the equation of motion here?