Final answer:
To ensure the solution of an initial value problem remains finite as t approaches infinity, the equation must feature stable equilibria or damping, and any exponential growth must be controlled or negated.
Step-by-step explanation:
The question is asking to find the value of y0 for which the solution of the initial value problem of a differential equation remains finite as t approaches infinity. The differential equation in question is not provided, so we will generalize the response.
For a solution to remain finite as t goes to infinity, the equation needs to have stable equilibria or be damped. This typically involves ensuring that any exponential terms within the solution have non-positive exponents, or that trigonometric function terms such as sine and cosine are bounded.
Because the question mentions sin(3m-1(0.00m)) = 0.00, we can infer the question pertains to wave-like solutions where an antinode is present. The antinode allows the solution at that point to oscillate with time. For a wave equation, staying finite typically means the amplitude does not grow with time. Thus, ensuring the amplitude remains constant or dissipative effects are present is necessary.
In physical terms, for a standing wave represented by the equation y(x, t) = A sin(kx + φ) cos(ωt), where φ is a phase constant, A is the amplitude, ω is the angular frequency of the temporal part, and k is the wavenumber, the amplitude A defines the maximum displacement, which should be a finite value for the solution to remain finite with time.