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Without using a mass matrix, ode solvers like ode45 can solve y'=f(t,y).

But there is an option of mass matrix in ode solvers for problems that involve a "mass" matrix, M(t,y)y'=f(t,y).

What exactly is the "mass" matrix? Does this term come from the mass of mass-spring system oscillation? I cannot find an example code about this in the documentation. Also, it seems that I can encode information about t and y in f(t,y) in the equation of y'=f(t,y). In what situation/example will M(t,y)y'=f(t,y) arise where M(t,y) is needed? 

The mass matrix is an important component in solving ordinary differential equations (ODEs) numerically using MATLAB's ODE solvers. In an ODE system of the form M(x) * x' = f(x,t), where M(x) is a matrix that can vary with respect to the independent variable x, and x is a vector of dependent variables, the mass matrix represents the system's inertia.

In MATLAB, the mass matrix can be specified as an input to ODE solvers such as ode15s and ode23s. When solving the ODE system, the solver requires the mass matrix to calculate the step size and ensure numerical stability.

If the mass matrix is constant and not a function of x, it can be specified as a matrix. However, if it is a function of x, it needs to be defined as a MATLAB function that takes x as input and returns the matrix.

Specifying the mass matrix in ODE solvers can improve the accuracy of the solution and reduce computational time. It is particularly useful when dealing with stiff systems, which are characterized by a large disparity in the time scales of the ODEs.

Overall, understanding the mass matrix and its role in ODE solvers is essential for effectively solving and simulating dynamic systems in MATLAB.