Differential equations are used in areas such as engineering, physics, chemistry, etc.Differential equations are the most prevalent mathematical models being employed by scientists. They are used to study the flows of fluids (water), heat (thermal convection) and mass (rho). Differential equations can be set up in several different forms that include differential, difference equation and difference-differential equation. The general idea that underlies all these types of differential equations is the concept of derivative which is a ratio between two functions (the rate of change) which will show how fast one function changes with respect to another function.The term 'beta' refers to an enhancement or increase in performance. In this context, it is used in the sense of The number of differential equations for which exact solutions have been found is surprisingly small. Only ~100 formulae are known to approximate a given differential equation. “Exact solutions” means that the solution exactly matches the solution of the differential equation with the same boundary conditions. The exact solutions may be expressed in terms of the functions, which are the solutions to the differential equation.The use of differential equations in science and engineering is one of the most important results in mathematical modeling since it enables modelling problems according to their real-life requirements and gives accurate and reliable predictions. Differential equations can be solved graphically (numerical solution) or analytically (theory). The analytical solution may be used for some specific types of problems where numerical methods are inappropriate. Theory is mainly concerned with the mathematical foundations if the differential equation, more precisely its theory of solution. The theory of solutions are based on basic principles that have been developed by mathematicians over many centuries. This theory of solution is used to find all solutions of differential equations that satisfy certain requirements. “Non-trivial” means that the number of unknowns is one more than the number of equations so there are at most two equations. The main purpose of differential equation is to model physical events, which can be natural phenomena or artificial devices, so it requires some knowledge about scientific laws and their application. For example, the motion of a planet around the sun has to obey Newton's law (the force exerted by the planet on the sun) and Kepler's laws (the forces exerted on a planet by other planets). Differential equations are used to study the changes in the density of a fluid flowing through a rectangular channel. The value of density varies with time. solve this problem by employing differential equations. The differential equation is where x is the distance measured along the channel, t is the time and p is the density of liquid at any instant. The boundary conditions are that liquid flows through one end of channel (x=0) and liquid comes out at other end (x = L). Substituting boundary conditions into Eqn.(1), we get This differential equation will be solved numerically using Runge-Kutta method which will be discussed later.
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