Some Nice Dynamics

Although this is a nonlinear equation, it models a particle of mass $1$ that is subject to a force $3-x^2$ (which is always directed towards the value $x=\sqrt{3}$ ) and a further damping force $-\dot{x}$ . Thus it is clear that the particle will (eventually) settle down to the equilibrium position of $x = \sqrt{3}$ . Since we are asked for a maximum value, and not a supremum value, the first swing of the oscillation must take the particle from $x=0$ to beyond $x=\sqrt{3}$ , and the maximum value of $x$ will then be obtained as the maximum value of $x$ during this first swing.

Numerical integration tells us that the maximum value occurs at $t = 2.10373$ , and is $\boxed{2.24513}$ .