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Differential Equations 10.11.06

Differential Equations 10.11.06 - D IFFERENTIAL EQUATIONS...

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He who seeks for methods without having a definite problem in mind seeks for the most part in vain. – D. HILBERT 9.1 Introduction In Class XI and in Chapter 5 of the present book, we discussed how to differentiate a given function f with respect to an independent variable, i.e., how to find f ( x ) for a given function f at each x in its domain of definition. Further, in the chapter on Integral Calculus, we discussed how to find a function f whose derivative is the function g , which may also be formulated as follows: For a given function g , find a function f such that dy dx = g ( x ), where y = f ( x ) ... (1) An equation of the form (1) is known as a differential equation . A formal definition will be given later. These equations arise in a variety of applications, may it be in Physics, Chemistry, Biology, Anthropology, Geology, Economics etc. Hence, an indepth study of differential equations has assumed prime importance in all modern scientific investigations. In this chapter, we will study some basic concepts related to differential equation, general and particular solutions of a differential equation, formation of differential equations, some methods to solve a first order - first degree differential equation and some applications of differential equations in different areas. 9.2 Basic Concepts We are already familiar with the equations of the type: x 2 – 3 x + 3 = 0 ... (1) sin x + cos x = 0 ... (2) x + y = 7 ... (3) Chapter 9 DIFFERENTIAL EQUATIONS Henri Poincare (1854-1912 )
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MATHEMATICS 380 Let us consider the equation: dy x y dx ± = 0 ... (4) We see that equations (1), (2) and (3) involve independent and/or dependent variable (variables) only but equation (4) involves variables as well as derivative of the dependent variable y with respect to the independent variable x . Such an equation is called a differential equation . In general, an equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation. A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation, e.g., 3 2 2 2 d y dy dx dx § · ± ¨ ¸ © ¹ = 0 is an ordinary differential equation .... (5) Of course, there are differential equations involving derivatives with respect to more than one independent variables, called partial differential equations but at this stage we shall confine ourselves to the study of ordinary differential equations only. Now onward, we will use the term ‘differential equation’ for ‘ordinary differential equation’. $ Note 1. We shall prefer to use the following notations for derivatives: 2 3 2 3 , , dy d y d y y y y dx dx dx c cc ccc 2. For derivatives of higher order, it will be inconvenient to use so many dashes as supersuffix therefore, we use the notation y n for n th order derivative n n d y dx .
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