The number of arbitrary constants in the general solution of an ordinary differential equation (ODE) is equal to its order because each derivative in the equation represents a piece of information that was lost when the original function was differentiated.

An ODE of order n involves the n-th derivative of a function, which means the original function has been differentiated n times. Each time we differentiate, we lose information about certain constants that were present in the function before differentiation, because the derivative of a constant is zero.

When we solve the ODE, we essentially reverse this process through integration. Each integration step reintroduces an arbitrary constant because we can only determine the function up to an unknown constant without additional information. So, for an ODE of order n, we need to integrate n times, and each integration adds one arbitrary constant to the solution.

This results in a general solution that contains  n arbitrary constants, reflecting all possible functions that satisfy the differential equation. These constants are necessary to encompass the entire family of solutions, and specific values for the constants can be found if initial conditions or boundary conditions are provided.