1. The problem asks for the governing equation of a microbeam. 2. Microbeams are small-scale beams where size effects become significant, often modeled using non-classical beam theories such as the strain gradient elasticity or couple stress theory. 3. The classical Euler-Bernoulli beam equation is $$EI\frac{\partial^4 w}{\partial x^4} = q(x,t)$$ where $w$ is the deflection, $E$ is Young's modulus, $I$ is the moment of inertia, and $q$ is the distributed load. 4. For microbeams, the governing equation is modified to include size-dependent effects. One common form using strain gradient theory is: $$EI\frac{\partial^4 w}{\partial x^4} - \mu \frac{\partial^6 w}{\partial x^6} = q(x,t)$$ where $\mu$ is a material length scale parameter capturing microstructure effects. 5. This higher-order differential equation accounts for additional stiffness due to microstructure, making it more accurate for micro-scale beams. 6. In summary, the governing equation of a microbeam is a modified Euler-Bernoulli beam equation with additional higher-order derivatives to capture size effects: $$EI\frac{\partial^4 w}{\partial x^4} - \mu \frac{\partial^6 w}{\partial x^6} = q(x,t)$$ This equation must be solved with appropriate boundary conditions depending on the beam setup.