How to fix the discrepancy in the results obtained by ode45 versus Newton-Raphson method?

Hello everyone,

I am trying to solve a 1D vibration problem: . Where .

With this problem, I have tried two ways:

1. I discretized the ϕ in an N-loop and update it after each time step . Inside this N-loop, I use ode45 to solve the following equation: .
2. I discretized the ϕ in an Newmark N-loop and update it after each time step . Inside this Newmark N-loop, I use Newton-Raphson to solve and update the above equation.

Theoretically, I expect there should be similar outputs on both of the above methods. But they are not. Is there any ideas why we have this discrepancy here? I am attaching the m-script here so that you can easily imagine the both algorithms and outputs.

Thanks in advance. Minh