Discrete differential equations are used to describe the effects of a change in initial conditions on changes in values of physical quantities. These equations can be formulated in terms of a set of first-order and second-order differential equations. Differential equations are particularly useful for describing the effects of a change in the initial conditions of a system. If the solution of a first-order differential equation is negative, then the value of a physical quantity is affected by the change in the system’s initial conditions; the equations that describe these changes can be used to describe the process of gravitational equilibrium.
Second-order differential equations describe the effects of a change in initial conditions on the values of physical quantities in a system. In second-order equations, only the first derivative, the integral of the first-order derivative, is involved. The second derivative of an initial condition, which is the difference between the initial condition and its value at a point in time, is called the second derivative. In the equations that describe the effects of gravity on a system, the second derivative is often ignored. A second-order differential equation, however, can be used to calculate the velocity change of a mass or fluid at some point in time.
Third-order differential equations describe the effects of a change in initial conditions on the values of physical quantities in a system. A third-order differential equation involves a single first-order equation and uses a power series to describe the change in values at different points in time. The expression for this power series includes the first derivative of the value of the quantity and the changes in time and velocity at those points in time. A third-order differential equation describes the effect of a change in a velocity function on the values of a number of physical quantities.
In the case of equations that describe the effects of gravity on a system, a derivative of the value of the velocity function is added to a first-order equation in the integral form to obtain the expression of the change in velocity function as a function of the change in time. Differential equations are sometimes used to describe the effects of the change in the initial conditions of a system on the value of a quantity at a point in time and its change in velocity; these equations are called the power equations.
A fourth-order differential equation includes a first-order equation that is modified in the integral form to include the changes in time and velocity at points in time and place. In such equations, the first derivative is added as a constant term and a second derivative, which are equal to the second derivative of the first derivative, is also added as a constant term, whereas in the integral form the first derivative is always zero; the results of these modifications will be the change in the values of the initial conditions of the system.
Fifth-order equations describe the effects of a change in the values of time and the velocities of particles and objects in a system, in terms of their change in position, at different times. Sixth-order equations describe the effects of these values of time and velocities on the motion of the particles and objects.
Seventh-order equations describe the effects of the velocities of objects and particles, in terms of their change in positions, on the changes of the values of their initial velocities. Eighth-order equations describe the changes in the values of the initial velocities of objects and particles and the changes in time and position. of the velocities of particles and objects in a system, in terms of their change in positions, at different times.
