Differential Equations Differential Equations Today I am going to tell you about a very interesting field of mathematics: differential equations. A differential equation is an equation for the functions which are unknown and are consisting of one or more variables. These variables generally relates the values of these unknown functions itself and its derivatives. These derivatives can be of various orders depending on the dependent and independent variables of the given equation. Differential Equations are used in various fields like physics, economics etc. These equations are used in real world applications. One of the examples is the determination of the velocity of a box which is falling through the air and we can only consider the resistance of air and the gravity. Now the acceleration the falling box towards the plane or ground is the acceleration due to the gravity minus the deceleration due to air resistance. Here the gravitation is assumed to be constant and the air resistance can be modeled as proportional to the velocity of the falling box. Know More About Chi Square Test Math.Tutorvista.com
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This indicates that the acceleration of the box is a derivative of velocity of the box. So the acceleration completely depends on the velocity of the box. Now if we find the velocity as a function of time then it will be known as differentiation and if we write this in terms then we will get a differential equation as: dv/dt = d^2x/dt dv/dt – d^2 /dt = 0 Where v is the velocity and x is the distance and t is time. And this is known as the differential equation. Now here are some types of these Differential Equations: 1. ODE: ODE stands for Ordinary Differential Equations. These equations are consisting of unknown function which is a function of a single variable that is independent. These functions can be of real or complex valued or generally found as vector valued or matrix valued. In short this is a system of ODE for a single function. 2. ODE’s classification: The ordinary differential equations are further divided into two categories which are: a. First order ODE b. Second order ODE These classifications of the ODEs are based on the order of the highest derivative of the dependent variable with respect to remaining independent variable that exists in the equation. Learn More :- Linear Regression
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3. PDE: PDE is basically stands for the Partial Differential Equation. In these equations the function is a function of many or multiple variables which are independent. The equation includes only its partial derivatives. Order of the PDEs is defined in the same manner as defined for the ODEs. For example: Inhomogeneous first order differential equation with unknown function f of x, and constants c and b: df/dx = c*f + x^2 And similarly the Homogeneous second order Ordinary differential equation: d^2 f/d x^2 – x(df/dx) + f = 0.
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