Introduction to differential equation

Introduction to differential equation

• 1.
  Introduction to DifferentialIntroductionto Differential EquationsEquations CSE:108 Bappa Sarkar Lecturer CSE, IU, Kushtia
• 2.
  Definition: A differential equationis an equation containing an unknown function and its derivatives. 32 += x dx dy 032 2 =++ ay dx dy dx yd 36 4 3 3 =+      + y dx dy dx yd Examples:. y is dependent variable and x is independent variable, and these are ordinary differential equations 1. 2. 3. Ordinary differential equations
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  Partial Differential Equation Examples: 02 2 2 2 = ∂ ∂ + ∂ ∂ y u x u 04 4 4 4 = ∂ ∂ + ∂ ∂ t u x u t u t u x u ∂ ∂ − ∂ ∂ = ∂ ∂ 2 2 2 2 uis dependent variable and x and y are independent variables, and is partial differential equation. u is dependent variable and x and t are independent variables 1. 2. 3.
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  Order of DifferentialEquation The order of the differential equation is order of the highest derivative in the differential equation. Differential Equation ORDER 32 += x dx dy 0932 2 =++ y dx dy dx yd 36 4 3 3 =+      + y dx dy dx yd 1 2 3
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  Degree of DifferentialEquation Differential Equation Degree 032 2 =++ ay dx dy dx yd 36 4 3 3 =+      + y dx dy dx yd 03 53 2 2 =+      +      dx dy dx yd 1 1 3 The degree of a differential equation is power of the highest order derivative term in the differential equation.
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  Linear Differential Equation Adifferential equation is linear, if 1. Dependent variable and its derivatives are of degree one, 2. Coefficients of a term does not depend upon dependent variable. Example: 36 4 3 3 =+      + y dx dy dx yd is non - linear because in 2nd term is not of degree one. .0932 2 =++ y dx dy dx ydExample: is linear. 1. 2.
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  Example: 3 2 2 2 x dx dy y dx yd x =+ is non- linear because in 2nd term coefficient depends on y. 3. Example: is non - linear because y dx dy sin= −+−= !3 sin 3 y yy is non – linear 4.
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  First Order OrdinaryDifferential equation 8
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  Second order OrdinaryDifferential Equation 9
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  nth – orderlinear differential equation 1. nth – order linear differential equation with constant coefficients. ( )xgya dx dy a dx yd a dx yd a dx yd a n n nn n n =+++++ − − − 012 2 21 1 1 .... 2. nth – order linear differential equation with variable coefficients ( ) ( ) ( ) ( ) ( ) ( )xgyxa dx dy xa dx yd xa dx yd xa dx dy xa n n nn =+++++ − − 012 2 2 1 1 ...... 10