In terms of mathematics, we say that the differential equation is the relationship that involves the derivative of a function or a dependent variable with respect to an independent variable. A differential equationis an equation which contains one or more terms which involve the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) dy/dx = f(x) Here “x” is an independent variable and “y” is a dependent variable For example, dy/dx = 5x A differential equation that contains derivatives which are either partial derivatives or ordinary derivatives. Equations that appear in applications tend to be second order, although higher order equations do appear from time to time. This second‐order linear differential equation with constant coefficients can be expressed in the more standard form . First Order Differential Equations In “real-world,” there are many physical quantities that can be represented by functions involving only one of the four variables e.g., (x, y, z, t) Equations involving highest order derivatives of order one = 1st order differential equations Examples: They can describe exponential growth and decay, the population growth of species or the change in investment return over time. Applications of Second‐Order Equations. The parameter that will arise from the solution of this first‐order differential equation will be determined by the initial condition v(0) = v 1 (since the sky diver's velocity is v 1 at the moment the parachute opens, and the “clock” is reset to t = 0 at this instant). Similarly, y’’, y’’’, …, so on, denoted nu… Hence, it is a generally assumed that the world is “second order… Application 1 : Exponential Growth - Population Let P(t) be a quantity that increases with time t and the rate of increase is proportional to the same quantity P as follows d P / d t = k P where d p / d t is the first derivative of P, k > 0 and t is the time. Differential equations have wide applications in various engineering and science disciplines. This separable equation is solved as follows: In general, modeling of the variation of a physical quantity, such as temperature,pressure,displacement,velocity,stress,strain,current,voltage,or concentrationofapollutant,withthechangeoftimeorlocation,orbothwould result in differential equations. The derivatives re… Differential equations have a remarkable ability to predict the world around us. The auxiliary polynomial equation is mr 2 + Kr + k = 0, whose roots are . They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. The solution to the above first order differential equation … It is represented as; Or y’=\frac{d(y)}{d(t)} Or f(x,y) = \frac{d(y)}{d(x)} = \frac{d(y)}{d(t)}= y’ Or x1\(\frac{d(y)}{d(x1)}\) + x2 \(\frac{d(y)}{d(x2)}\) = y Where x is the independent variable And y is the dependent variable, as its function is dependent on the values of x. Y’ denotes one derivative.

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