This question is incomplete. We are only given one initial condition, but for a second order differential equation, we'd need two initial conditions to completely determine the behaviour of the system. So we can't determine the maximum speed. If they'd told us something like the particle was initially at rest, then we could have.
This would require the assumption that the initial velocity is 0 (i.e. it starts at rest), which is not given to us.
Like imagine releasing a pendulum (this is approximately a simple harmonic oscillator for small angles of amplitude). Its angular frequency is inherent (based on the length of the string and the gravitational acceleration on Earth). In our problem, it's like we are given the angular frequency (7) of the pendulum, and also where we started it. But we're not told how fast we started it. We could have just released it from the given starting position, or given it a bit of a push etc. This initial velocity would affect the amplitude, and is thus the extra piece of initial condition info we need to fully determine the behaviour of the system.