# Differential Equations Solutions

Differential Equations Solutions are one of the mathematical tools used to analyze Vibration and Electrical Circuits. These Two types of Problems require a second Ordinary Differential Equation (ODE) Solutions. Mechanical Vibration problems have a single degree of freedom system with or without dampening. Systems include internal forces generated by the mass (m) times acceleration, dampening (c) coefficient times velocity and the stiffness of the system (k) times the displacement. this equation can be formulated by ordinary physics and Newtons second law F = Ma, Impulse, and Momentum, D’Albert’s principal with the application of Virtual Work which is based on dynamic equilibrium. Or setting up the free body diagram and maintaining static equilibrium.

The differential equation of motion without a forcing function and damping yields the system’s circular frequency;

\[\omega = \sqrt{K/M} , f = \frac{\omega }{2\pi } = \frac{1}{T} ;\\Ma+Cv+Kx =F(t) ; \\ M\ddot{x} + C\dot{x}+ Kx = F(t)\]

If we look closer we can see that the fundamental