The fundamental mathematical principle for linear systems, powerfully demonstrated by simulating the Damped Wave Equation with fixed endpoints, is that homogeneity dictates the trivial solution. Specifically, when the PDE itself and the Boundary Conditions are homogeneous (equal to zero), the physical system requires an inhomogeneous Initial Condition (either non-zero initial displacement, $u(x, 0) \neq 0$, or velocity, $u_t(x, 0) \neq 0$ ) to introduce the necessary energy for motion. The non-trivial simulation, starting with a plucked string, shows energy propagating as waves before decaying due to damping; conversely, the trivial simulation, starting perfectly still, remains motionless ( $u(x, t)=0$ ), confirming that without an initial input, a stable, homogeneous system cannot spontaneously generate a measurable solution.
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