Page 1 DIFFERENTIAL EQUATIONS 379 ? He who seeks for methods without having a definite problem in mind seeks for the most part in vain. â€“ 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 dif ferential 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 â€“ 3x + 3 = 0 ... (1) sin x + cos x = 0 ... (2) x + y = 7 ... (3) Chapter 9 DIFFERENTIAL EQUATIONS Henri Poincare (1854-1912 ) 2019-20 Page 2 DIFFERENTIAL EQUATIONS 379 ? He who seeks for methods without having a definite problem in mind seeks for the most part in vain. â€“ 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 dif ferential 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 â€“ 3x + 3 = 0 ... (1) sin x + cos x = 0 ... (2) x + y = 7 ... (3) Chapter 9 DIFFERENTIAL EQUATIONS Henri Poincare (1854-1912 ) 2019-20 MATHEMATICS 380 Let us consider the equation: dy xy 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 ?? + ?? ?? = 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 â€˜differential equationâ€™ for â€˜ordinary differential equationâ€™. ? Note 1. We shall prefer to use the following notations for derivatives: 2 3 2 3 ,, dy d y d y y yy dx dx dx ' '' ''' = == 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 nth order derivative n n dy dx . 9.2.1. Order of a differential equation Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Consider the following differential equations: dy dx = e x ... (6) 2019-20 Page 3 DIFFERENTIAL EQUATIONS 379 ? He who seeks for methods without having a definite problem in mind seeks for the most part in vain. â€“ 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 dif ferential 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 â€“ 3x + 3 = 0 ... (1) sin x + cos x = 0 ... (2) x + y = 7 ... (3) Chapter 9 DIFFERENTIAL EQUATIONS Henri Poincare (1854-1912 ) 2019-20 MATHEMATICS 380 Let us consider the equation: dy xy 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 ?? + ?? ?? = 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 â€˜differential equationâ€™ for â€˜ordinary differential equationâ€™. ? Note 1. We shall prefer to use the following notations for derivatives: 2 3 2 3 ,, dy d y d y y yy dx dx dx ' '' ''' = == 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 nth order derivative n n dy dx . 9.2.1. Order of a differential equation Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Consider the following differential equations: dy dx = e x ... (6) 2019-20 DIFFERENTIAL EQUATIONS 381 2 2 dy y dx + = 0 ... (7) 3 3 2 2 3 2 dy d y x dx dx ?? ? ? + ?? ? ? ?? ?? = 0 ... (8) The equations (6), (7) and (8) involve the highest derivative of first, second and third order respectively. Therefore, the order of these equations are 1, 2 and 3 respectively . 9.2.2 Degree of a differential equation To study the degree of a differential equation, the key point is that the differential equation must be a polynomial equation in derivatives, i.e., y', y?, y?' etc. Consider the following differential equations: 2 32 32 2 d y d y dy y dx dx dx ?? + -+ ?? ?? = 0 ... (9) 2 2 sin dy dy y dx dx ?? ?? +- ?? ?? ?? ?? = 0 ... (10) sin dy dy dx dx ?? + ?? ?? = 0 ... (11) We observe that equation (9) is a polynomial equation in y?', y? and y', equation (10) is a polynomial equation in y' (not a polynomial in y though). Degree of such differential equations can be defined. But equation (11) is not a polynomial equation in y' and degree of such a differential equation can not be defined. By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation. In view of the above definition, one may observe that differential equations (6), (7), (8) and (9) each are of degree one, equation (10) is of degree two while the degree of differential equation (11) is not defined. ? Note Order and degree (if defined) of a differential equation are always positive integers. 2019-20 Page 4 DIFFERENTIAL EQUATIONS 379 ? He who seeks for methods without having a definite problem in mind seeks for the most part in vain. â€“ 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 dif ferential 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 â€“ 3x + 3 = 0 ... (1) sin x + cos x = 0 ... (2) x + y = 7 ... (3) Chapter 9 DIFFERENTIAL EQUATIONS Henri Poincare (1854-1912 ) 2019-20 MATHEMATICS 380 Let us consider the equation: dy xy 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 ?? + ?? ?? = 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 â€˜differential equationâ€™ for â€˜ordinary differential equationâ€™. ? Note 1. We shall prefer to use the following notations for derivatives: 2 3 2 3 ,, dy d y d y y yy dx dx dx ' '' ''' = == 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 nth order derivative n n dy dx . 9.2.1. Order of a differential equation Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Consider the following differential equations: dy dx = e x ... (6) 2019-20 DIFFERENTIAL EQUATIONS 381 2 2 dy y dx + = 0 ... (7) 3 3 2 2 3 2 dy d y x dx dx ?? ? ? + ?? ? ? ?? ?? = 0 ... (8) The equations (6), (7) and (8) involve the highest derivative of first, second and third order respectively. Therefore, the order of these equations are 1, 2 and 3 respectively . 9.2.2 Degree of a differential equation To study the degree of a differential equation, the key point is that the differential equation must be a polynomial equation in derivatives, i.e., y', y?, y?' etc. Consider the following differential equations: 2 32 32 2 d y d y dy y dx dx dx ?? + -+ ?? ?? = 0 ... (9) 2 2 sin dy dy y dx dx ?? ?? +- ?? ?? ?? ?? = 0 ... (10) sin dy dy dx dx ?? + ?? ?? = 0 ... (11) We observe that equation (9) is a polynomial equation in y?', y? and y', equation (10) is a polynomial equation in y' (not a polynomial in y though). Degree of such differential equations can be defined. But equation (11) is not a polynomial equation in y' and degree of such a differential equation can not be defined. By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation. In view of the above definition, one may observe that differential equations (6), (7), (8) and (9) each are of degree one, equation (10) is of degree two while the degree of differential equation (11) is not defined. ? Note Order and degree (if defined) of a differential equation are always positive integers. 2019-20 MATHEMATICS 382 Example 1 Find the order and degree, if defined, of each of the following differential equations: (i) cos 0 dy x dx -= (ii) 2 2 2 0 d y dy dy xy x y dx dx dx ?? + -= ?? ?? (iii) 2 0 y y ye ' ''' ++ = Solution (i) The highest order derivative present in the differential equation is dy dx , so its order is one. It is a polynomial equation in y' and the highest power raised to dy dx is one, so its degree is one. (ii) The highest order derivative present in the given differential equation is 2 2 dy dx , so its order is two. It is a polynomial equation in 2 2 dy dx and dy dx and the highest power raised to 2 2 dy dx is one, so its degree is one. (iii) The highest order derivative present in the differential equation is y''' , so its order is three. The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined. EXERCISE 9.1 Determine order and degree (if defined) of differential equations given in Exercises 1 to 10. 1. 4 4 sin( ) 0 dy y dx ''' += 2. y' + 5y = 0 3. 4 2 2 30 ds d s s dt dt ?? += ?? ?? 4. 2 2 2 cos 0 d y dy dx dx ?? ?? += ? ? ?? ?? ?? 5. 2 2 cos3 sin3 dy xx dx =+ 6. 2 () y''' + (y?) 3 + (y') 4 + y 5 = 0 7. y''' + 2y? + y' = 0 2019-20 Page 5 DIFFERENTIAL EQUATIONS 379 ? He who seeks for methods without having a definite problem in mind seeks for the most part in vain. â€“ 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 dif ferential 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 â€“ 3x + 3 = 0 ... (1) sin x + cos x = 0 ... (2) x + y = 7 ... (3) Chapter 9 DIFFERENTIAL EQUATIONS Henri Poincare (1854-1912 ) 2019-20 MATHEMATICS 380 Let us consider the equation: dy xy 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 ?? + ?? ?? = 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 â€˜differential equationâ€™ for â€˜ordinary differential equationâ€™. ? Note 1. We shall prefer to use the following notations for derivatives: 2 3 2 3 ,, dy d y d y y yy dx dx dx ' '' ''' = == 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 nth order derivative n n dy dx . 9.2.1. Order of a differential equation Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Consider the following differential equations: dy dx = e x ... (6) 2019-20 DIFFERENTIAL EQUATIONS 381 2 2 dy y dx + = 0 ... (7) 3 3 2 2 3 2 dy d y x dx dx ?? ? ? + ?? ? ? ?? ?? = 0 ... (8) The equations (6), (7) and (8) involve the highest derivative of first, second and third order respectively. Therefore, the order of these equations are 1, 2 and 3 respectively . 9.2.2 Degree of a differential equation To study the degree of a differential equation, the key point is that the differential equation must be a polynomial equation in derivatives, i.e., y', y?, y?' etc. Consider the following differential equations: 2 32 32 2 d y d y dy y dx dx dx ?? + -+ ?? ?? = 0 ... (9) 2 2 sin dy dy y dx dx ?? ?? +- ?? ?? ?? ?? = 0 ... (10) sin dy dy dx dx ?? + ?? ?? = 0 ... (11) We observe that equation (9) is a polynomial equation in y?', y? and y', equation (10) is a polynomial equation in y' (not a polynomial in y though). Degree of such differential equations can be defined. But equation (11) is not a polynomial equation in y' and degree of such a differential equation can not be defined. By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation. In view of the above definition, one may observe that differential equations (6), (7), (8) and (9) each are of degree one, equation (10) is of degree two while the degree of differential equation (11) is not defined. ? Note Order and degree (if defined) of a differential equation are always positive integers. 2019-20 MATHEMATICS 382 Example 1 Find the order and degree, if defined, of each of the following differential equations: (i) cos 0 dy x dx -= (ii) 2 2 2 0 d y dy dy xy x y dx dx dx ?? + -= ?? ?? (iii) 2 0 y y ye ' ''' ++ = Solution (i) The highest order derivative present in the differential equation is dy dx , so its order is one. It is a polynomial equation in y' and the highest power raised to dy dx is one, so its degree is one. (ii) The highest order derivative present in the given differential equation is 2 2 dy dx , so its order is two. It is a polynomial equation in 2 2 dy dx and dy dx and the highest power raised to 2 2 dy dx is one, so its degree is one. (iii) The highest order derivative present in the differential equation is y''' , so its order is three. The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined. EXERCISE 9.1 Determine order and degree (if defined) of differential equations given in Exercises 1 to 10. 1. 4 4 sin( ) 0 dy y dx ''' += 2. y' + 5y = 0 3. 4 2 2 30 ds d s s dt dt ?? += ?? ?? 4. 2 2 2 cos 0 d y dy dx dx ?? ?? += ? ? ?? ?? ?? 5. 2 2 cos3 sin3 dy xx dx =+ 6. 2 () y''' + (y?) 3 + (y') 4 + y 5 = 0 7. y''' + 2y? + y' = 0 2019-20 DIFFERENTIAL EQUATIONS 383 8. y' + y = e x 9. y? + (y') 2 + 2y = 0 10. y? + 2y' + sin y = 0 11. The degree of the differential equation 3 2 2 2 sin 1 0 d y dy dy dx dx dx ?? ?? ?? + + += ? ? ?? ?? ?? ?? ?? is (A) 3 (B) 2 (C) 1 (D) not defined 12. The order of the differential equation 2 2 2 23 0 d y dy xy dx dx - += is (A) 2 (B) 1 (C) 0 (D) not defined 9.3. General and Particular Solutions of a Differential Equation In earlier Classes, we have solved the equations of the type: x 2 + 1 = 0 ... (1) sin 2 x â€“ cos x = 0 ... (2) Solution of equations (1) and (2) are numbers, real or complex, that will satisfy the given equation i.e., when that number is substituted for the unknown x in the given equation, L.H.S. becomes equal to the R.H.S.. Now consider the differential equation 2 2 0 dy y dx += ... (3) In contrast to the first two equations, the solution of this differential equation is a function f that will satisfy it i.e., when the function f is substituted for the unknown y (dependent variable) in the given differential equation, L.H.S. becomes equal to R.H.S.. The curve y = f (x) is called the solution curve (integral curve) of the given differential equation. Consider the function given by y = f (x) = a sin (x + b), ... (4) where a, b ? R. When this function and its derivative are substituted in equation (3), L.H.S. = R.H.S.. So it is a solution of the differential equation (3). Let a and b be given some particular values say a = 2 and 4 b p = , then we get a function y = f 1 (x) = 2sin 4 x p ?? + ?? ?? ... (5) When this function and its derivative are substituted in equation (3) again L.H.S. = R.H.S.. Therefore f 1 is also a solution of equation (3). 2019-20Read More

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