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Differential Equations.pptx
- 1. Lets Fall InLove with Mathematics! Mathematics Differential Equation ANAND KUMAR
- 2. index What will welearn today? Index Differential Equations – Definition Order and Degree Of a D.E Formation Of a D.E
- 3. 26-11-2022 3 Introduction An equationinvolving independent variable (x), dependent variable (y), and the derivatives of dependent variables with respect independent variables is called a differential equation. e.g. (i) (ii) dy = cos x dx (iii) y = x + a (iv) dy x ln x dx dy dx 2 4 2 2 dy d y 1 b dx dx Differential Equations
- 4. 26-11-2022 4 Differential Equations OrderAnd Degree Of A Differential Equation Order : The order of a differential equation is the order of the derivative of the highest order occurring in the differential equation. Degree : The degree of a differential equation is the degree of the highest order differential coefficient appearing in it, provided it can be expressed as a polynomial equation in derivatives. e.g. dy xy sin x dx
- 5. 26-11-2022 5 Find theorder and degree of the following differential equations (i) (ii) (iii) 5 6 2 y 2 y 3x y 0 y sin(y ) 2 3 2 d y dy 3 dx dx Differential Equations
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- 7. 26-11-2022 7 (iv) (v) Differential Equations 5 4 23 2 d y dy 1 dx dx 2 2 d y dy = x ln dx dx
- 8. 26-11-2022 8 Differential Equations FormationOf A Differential Equation Differential equation corresponding to a family of curve will have : (a) Order exactly same as number of essential arbitrary constants in the equation of curve. (b) No arbitrary constant present in it.
- 9. 26-11-2022 9 Differential Equations Thedifferential equation corresponding to a family of curve can be obtained by using the following steps: (i) Identify the number of essential arbitrary constants in equation of curve. NOTE : If arbitrary constants appear in addition, subtraction, multiplication or division, then we can club them to reduce into one new arbitrary constant. (ii) Differentiate the equation of curve till the required order. (iii) Eliminate the arbitrary constant from the equation of curve and additional equations obtained in step (ii) above.
- 10. 26-11-2022 10 Differential Equations Example# Form a differential equation of family of straight lines passing through origin. 𝑌 𝑋 𝑂
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- 12. 26-11-2022 12 Differential Equations 𝑌 𝑋 𝑂 Example# Form a differential equation of family of circles touching x-axis at the origin
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- 14. 26-11-2022 14 Differential Equations 𝑌 𝑋 𝑂 Example# Obtain a differential equation of the family of curves y = a sin (bx + c) where a and c being arbitrary constant.
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- 17. 26-11-2022 17 Differential Equations SolutionOf A Differential Equation A solution of a differential equation is an equation which contains arbitrary constants as many as the order of the differential equation and is called the GENERAL SOLUTION(OR COMPLETE SOLUTION). Other solutions, obtained by giving particular values to the arbitrary constants in the general solution, are called PARTICULAR SOLUTIONS. Thus we see that the general solution of a differential equation of the nth order must contain n and only n independent arbitrary constants.
- 18. 26-11-2022 18 Differential Equations Example# Show that the function is a solution of the differential equation xy' + 2(1 + x2) y = 0. 2 2 -x 2x y+e =0
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- 20. 20 Differential Equations Solution OfA Differential Equation By The Method Of Variable Separation If the coefficient of dx is only a function of x and that of dy is only a function of y in the given differential equation, then the equation can be solved using variable separation method. Thus the general form of such an equation is Type I : The general solution of above equation is given by f(x)dx + φ(y).dy=C f(x)dx +φ(y)dy=0
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- 25. 26-11-2022 25 Differential Equations TypeII : , b ≠ 0 dy f (ax by c) dx To sole this equation substitute ax + by + c = t . Then the equation reduces to variable separable form. Example #
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- 30. Differential Equations Homogeneous DifferentialEquation A function f(x, y) is said to be a homogeneous function of its variables of degree n if the identity f(tx, ty) = tn f(x,y) is valid. For example, the function f(x, y) = x2 + y2 – xy is a homogeneous function of the second degree. A differential equation of the form , where f(x, y) is a homogenous polynomial of degree zero is called a homogenous differential equation. dy =f(x,y) dx Such equations are solved by substituting v = y/x (or x/y) and then separating the variables.
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- 39. 26-11-2022 39 Differential Equations EquationsReducible to Homogeneous Form Equation of the form (aB ≠ Ab) can be reduced to a homogenous form by changing the variables x, y to X, Y by equations x = X + h, y = Y + k, where h, k are constants to be chosen so as to make the given equation homogenous. dy ax by c dx Ax By C
- 40. 40 Differential Equations Example # (x+ 2y – 3)dy = (2x – y + 1)dx
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- 42. The general lineardifferential equation of order n is of the form where a0, a1, a2,..., an, f are either constants or functions of x alone. 26-11-2022 42 Differential Equations Linear Differential Equations The linear differential equations are those in which the dependent variable and its derivative appear only in their first degree and are not multiplied together. The coefficients may be constants or functions of x alone.
- 43. 26-11-2022 43 Differential Equations Note: Thata linear differential equation is always of the first degree but every differential equation of the first degree need not be linear. For example, the differential equation is not linear, though its degree is 1.
- 44. 44 Differential Equations The lineardifferential equation of the first order is of the form ....(1) where P and Q are constants or functions of x alone. dy Py Q dx
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- 58. 26-11-2022 58 Differential Equations DifferentialEquations Reducible to Linear Form An equation of the form where P and Q are constants or functions of x alone and n is constant except 0 and 1, is called a Bernoulli’s equation. n dy Py Qy dx Differential Equations
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- 70. 26-11-2022 70 Differential Equations Ifp and q are order and degree of differential equation y2 + 3x + x2y2 = sin x, then : (A) p > q (B) 2p = q (C) p = q (D) p < q 2 2 2 d y dx 1/3 dy dx Differential Equations
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- 72. 26-11-2022 72 Differential Equations Theorder of the differential equation whose general solution is given by y = (C1 + C2) sin (x + C3) – C4 is (A) 5 (B) 4 (C) 2 (D) 3 5 x C e Differential Equations
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- 74. 26-11-2022 74 Differential Equations Familyy = Ax + A3 of curve represented by the differential equation of degree (A) three (B) two (C) one (D) none of these Differential Equations
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- 76. 26-11-2022 76 Differential Equations If= e – 2y and y = 0 when x = 5, the value of x for y = 3 is (A) e5 (B) e6 + 1 (C) (D) loge 6 dy dx 6 9 2 e Differential Equations
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- 78. 26-11-2022 78 Differential Equations Solutionof differential equation xdy – y dx = 0 represents : (A) rectangular hyperbola (B) straight line passing through origin (C) parabola whose vertex is at origin (D) circle whose centre is at origin Differential Equations
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- 80. 26-11-2022 80 Differential Equations Thedegree and order of the differential equation of the family of all parabolas whose axis is x-axis, are respectively (1) 2, 1 (2) 1, 2 (3) 3, 2 (4) 2, 3 Differential Equations
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- 82. 26-11-2022 82 Differential Equations Thesolution of the differential equation satisfying the condition y (1) = 1 is (1) y = log x + x (2) y = x log x + x2 (3) y = xe(x – 1) (4) y = x log x + x dy x y dx x Differential Equations
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- 84. 26-11-2022 84 Differential Equations Thedifferential equation which represents the family of curves y = c1 , where c1 and c2 are arbitrary constants is (1) y = y2 (2) y = y y (3) y.y = y (4) y.y = (y)2 2 c x e Differential Equations
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- 86. 26-11-2022 86 Differential Equations Thesolution of the equation, sin 1 x + y is : (A) tan (x + y) + sec (x + y) = x + c (B) tan (x + y) - sec (x + y) = x + c (C) tan (x + y) + sec (x + y) - x + c = 0 (D) none dy dx Differential Equations
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- 89. 26-11-2022 89 Differential Equations Thesolution of the differential equation, = x2, where y(1) = 1 is (1) 4y = x4 + 3 (2) 4xy = x4 + 3 (3) 4y = x3 + 3 (4) 4xy = x3 + 3 dy y dx x Differential Equations
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- 91. 26-11-2022 91 Differential Equations Thesolution of the differential equation, (x + 2y3) = y is : (A) = y + c (B) = y2 + c (C) = y2 + c(D) = x2 + c dy dx 2 x y x y 2 x y y x Differential Equations
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- 93. Mathematics Next Class WillBe On ?
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