Understanding Differential Equations: A Comprehensive Overview
Explore the world of differential equations with this in-depth guide covering definitions, types, order classification, and examples. Discover how these equations play a crucial role in various disciplines like engineering, physics, economics, and biology.
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Presentation Transcript
Advance Calculus For Engineering By Prof. dr salah Raza saeed
Differential Equations Differential Equations https://scholar.google.com/citations?user=gBsVwVYAAAAJ&hl= en November 16, 2023 2
Contents Differential Equations Definition, order and degree First Order Differential Equations Integrable .... Separable ........ . Integrating Factor .... . Second Order Differential Equations Homogeneous . Inhomogeneous . Example Questions . . .
What is differential equations In mathematics a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Differential equations came into existence with the invention of calculus by Isaac Newton and Gottfried Leibniz. Types In general differential equations can be divided into several types. ordinary de partial de linear or non-linear de, and homogeneous or heterogeneous de.
Order of Differential Equation Differential Equations are classified on the basis of the order. The order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation. Example (i): In this equation, the order of the highest derivative is 3 hence, this is a third order differential equation. Example (ii): This equation represents a second order differential equation. This way we can have higher order differential equations i.e., nthorder differential equations. https://scholar.google.com/citations?user=gBsVwVYAAAAJ&hl= en November 16, 2023 5
First order differential equation First order differential equation The order of highest derivative in the case of first order differential equations is 1. A linear differential equation has order 1. In the case of linear differential equations, the first derivative is the highest order derivative. P and Q are either constants or functions of the independent variable only. This represents a linear differential equation whose order is 1. Second Order Differential Equation Second Order Differential Equation When the order of the highest derivative present is 2, then it is a second order differential equation. Example: Example: In this example, the order of the highest derivative is 2. Therefore, it is a second order differential equation.
Degree of Differential Equation The degree of the differential equation is represented by the power of the highest order derivative in the given differential equation. The differential equation must be a polynomial equation in derivatives for the degree to be defined. he order of this equation is 3 and the degree is 2 as the highest derivative is of order 3 and the exponent raised to the highest derivative is 2. Here, the exponent of the highest order derivative is one and the given differential equation is a polynomial equation in derivatives. Hence, the degree of this equation is 1.
Special Case (Degree is Not Defined) Suppose in a differential equation dy/dx = tan (x + y), the degree is 1, whereas for a differential equation tan (dy/dx) = x + y, the degree is not defined. These types of differential equations can be observed with other trigonometry functions such as sine, cosine and so on.
Ordinary differential equation An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable. Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. The general first-order differential equation for the function y = y(x) is written as dy /dx = f(x, y), where f(x, y) can be any function of the independent variable x and the dependent variable y.
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