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Definitions |
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Differential Equation: A differential equation is a relation between the independent, dependent variables and their differential coefficients. |
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Order of Differential Equation: The order of differential equation is defined to be the order of the highest order derivative of the dependent variable occurring in the differential equation. |
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Degree of a Differential Equation: The degree of a differential equation is the highest power of the highest order derivative after making the equation free from radicals and fractional indices as far as the derivatives are concerned. |
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Formation of a Differential Equation |
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The equation y = mx has one arbitrary constant and its differential equation is of order 1. The equation y = mx + c has two arbitrary constants and its differential equation is of order 2. |
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In general, if an equation contains n arbitrary constants, then we obtain its differential equation which is of order n, after eliminatory all the n constants. |
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The formation of a differential equation may be done by differentiating and eliminating arbitrary constants from the given equation. |
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Solution of a Differential Equation |
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Solution of a Differential Equation: The functional relation-ship between the independent variable and the dependent variable (such as y = f(x)) which satisfies the given differential equation is called the solution of the differential equation. |
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Particular solution of a differential equation: A solution obtained, by assigning particular values to the arbitrary constants in the general solution of the differential equation, is called its particular solution. |
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General solution of a differential equation: If the solution of a differential equation of order n contains n arbitrary constants, then it is called the General solution of the differential equation. |
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Classification of Differential Equation |
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Differential equation are classified according to their order and they are:
First Order Differential Equation, Higher Order Differential Equation, Linear Differential Equation, Non-linear differential equation. |
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An Alternative Form of a First-order First-degree Differential Equation |
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The fact that, Dy
- f(x, y) Dx ® 0 as
Dx ® 0 is expressed by
writing dy = f(x,y)dx is an alternative form of the differential equation |
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Some Methods of Solving First Order First Degree Differential Equation |
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The different ways of solving differential equation are a follows: (a). Method of separation of variables, (b). Homogeneous differential equations, (c). Linear differential equations. |
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Important points: (a). Degree and order of a differential equation, (b). How to form a differential equation, (c). Types of differential equations and their general solutions. |
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Special types of a Second Order Differential Equation |
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Second-order differential equations, by definition, contain a second
derivative, like d2y/dx2, for example. As well as the
second derivative, there may also be a first derivative in the equation and
sometimes a term involving just y itself. |
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The second-order differential equations we will look at have "constant coefficients", which means that these three possible terms, that is the second derivative, the first derivative and the y-term, all just have a number in front of them (their coefficient), rather than a function of x. |
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Applications |
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1. A wet porous substance in the open air loses its moisture at the rate proportional to the moisture content. If a sheet hung in the wind loses half of its moisture during the first hour, when will it have lost? |
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(i) 95% moisture, weather condition remaining constant |
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(ii) 90% moisture, weather condition remaining the same. |
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2. The decay rate of radium at any point t is proportional to its mass at that time. The mass is M0 at time t = 0. Form the differential equation and find the time when the mass will be halved. |
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Summary |
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1. An equation involving independent variables dependent variable and their derivatives is called a differential equation. |
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2. A differential equation which involves only one independent variable is called an ordinary differential equation. |
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3. To form a differential equation from a given equation in x, y and containing arbitrary constants. The given equation is differentiated successively as many times as the number of arbitrary constants. These equations are used to eliminate the arbitrary constants and the equation obtained is the required differential equation. |
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Conclusion |
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In this chapter we have the formation of differential equations and also some methods of solving the differential equations namely, variables separable methods, homogeneous differential equations and linear differential equations.
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