Question

The number of arbitrary constants in the solution of a differential equation of degree 2 and order 3 is

Answer 1

In general, the number of arbitrary constants of an ordinary differential equation (ODE) is given by the order of the highest derivative.
e.g.

y′=f(x)y′=f(x)

has oneone constant of integration,

y′′+y′=f(x)y″+y′=f(x)

has twotwo,

y′′′+y′′=f(x)y‴+y″=f(x)

has three,three, etc..
To generalise, suppose we've an ODE of the form

any(n)+an−1y(n−1)+⋯+a2y′′+a1y′+a0y=f(x),any(n)+an−1y(n−1)+⋯+a2y″+a1y′+a0y=f(x),

where aiai is a function of xx.
Then the number of arbitrary constants in the general solution to this equation