In a mechanical system, the equation (m\frac{d^2x}{dt^2} + kx = 0) describes the motion of a mass (m) attached to a spring with spring constant (k). What is the natural frequency of the system?

Consider the differential equation (y'' + 4y = \sin(2t)). What is the general solution to this equation?

In an electrical circuit, the equation (L\frac{di}{dt} + Ri = E) describes the current (i) flowing through an inductor with inductance (L), a resistor with resistance (R), and a voltage source (E). What is the time constant of the circuit?

A population of bacteria grows according to the differential equation (\frac{dN}{dt} = kN), where (N) is the population size and (k) is a constant. If the initial population size is (N_0), what is the population size at time (t)?

In a chemical reaction, the rate of change of the concentration of a reactant (A) is given by the differential equation (\frac{d[A]}{dt} = -k[A]^2), where (k) is a constant. What is the order of the reaction?

A spring-mass system is described by the differential equation (m\frac{d^2x}{dt^2} + kx = F_0\sin(\omega t)), where (m) is the mass, (k) is the spring constant, (F_0) is the amplitude of the applied force, and (\omega) is the angular frequency. What is the steady-state solution for the displacement (x)?

In a heat transfer problem, the temperature (u(x, t)) satisfies the partial differential equation (\frac{\partial u}{\partial t} = \alpha\frac{\partial^2 u}{\partial x^2}), where (\alpha) is a constant. What is the method of solution called?

A vibrating string is described by the wave equation (\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}), where (c) is the wave speed. What is the general solution to this equation?

In a fluid flow problem, the velocity field (\mathbf{u}(x, y, t)) satisfies the Navier-Stokes equations. What is the mathematical form of the Navier-Stokes equations?

In a mass-spring-damper system, the equation (m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t)) describes the displacement (x) of the mass. What is the damping ratio of the system?

A vibrating membrane is described by the partial differential equation (\frac{\partial^2 u}{\partial t^2} = c^2\nabla^2 u), where (c) is a constant. What is the method of solution called?

In a heat transfer problem, the temperature (u(x, y, z, t)) satisfies the partial differential equation (\frac{\partial u}{\partial t} = \alpha\nabla^2 u), where (\alpha) is a constant. What is the method of solution called?

A vibrating string is described by the wave equation (\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}), where (c) is the wave speed. What is the general solution to this equation?

In a fluid flow problem, the velocity field (\mathbf{u}(x, y, z, t)) satisfies the Navier-Stokes equations. What is the mathematical form of the Navier-Stokes equations?