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Introduction to Differential Equations 2014 ordinary differential equations Definition: A differential equation is an equation containing an unknown function and its derivatives. Examples:. 1. 2. dy 2x 3 dx d2y dy 3 ay 0 2 dx dx 4 d y dy 6y 3 3 dx dx y is dependent variable and x is independent variable, and these are ordinary differential equations 3 3. Partial Differential Equation Examples: 1. 2u 2u 2 0 2 x y u is dependent variable and x and y are independent variables, and is partial differential equation. 2. 4u 4u 4 0 4 x t 3. 2 u 2 u u 2 2 t x t u is dependent variable and x and t are independent variables Order of Differential Equation The order of the differential equation is order of the highest derivative in the differential equation. Differential Equation dy 2x 3 dx d2y dy 3 9y 0 2 dx dx 4 d 3 y dy 6y 3 3 dx dx ORDER 1 2 3 Degree of Differential Equation The degree of a differential equation is power of the highest order derivative term in the differential equation. Differential Equation d2y dy 3 ay 0 2 dx dx Degree 1 4 d 3 y dy 6y 3 3 dx dx 3 d 2 y dy 2 3 0 dx dx 1 5 3 Linear Differential Equation A differential 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: d2y dy 3 9 y 0. 1. 2 dx dx is linear. Example: 2. 4 d 3 y dy 6y 3 3 dx dx is non - linear because in 2nd term is not of degree one. Example: 3. d2y dy 3 x y x dx dx 2 2 is non - linear because in 2nd term coefficient depends on y. Example: 4. dy sin y dx is non - linear because y3 sin y y 3! is non – linear It is Ordinary/partial Differential equation of order… and of degree…, it is linear / non linear, with independent variable…, and dependent variable…. 1st – order 1. Derivative form: differential equation dy a1 x a 0 x y g x dx 2. Differential form: 1 xdy ydx 0 . 3. General form: dy f ( x, y ) dx or Differential Equation Chapter 1 dy f ( x, y, ) 0. dx 9 First Order Ordinary Differential equation Differential Equation Chapter 1 10 Second order Ordinary Differential Equation Differential Equation Chapter 1 11 –linear order linearequation differential 1. nth –nth order differential with constant coefficients. equation dny d n 1 y d2y dy a n n a n 1 n 1 .... a 2 2 a1 a0 y g x dx dx dx dx 2. nth – order linear differential equation with variable coefficients dy d n1 y d2y dy a n x a n1 x ...... a x a x a0 x y g x 2 1 n 2 dx dx dx dx Differential Equation Chapter 1 12 Solution of Differential Equation 2014 Differential Equation Chapter 1 13 Examples y=3x+c , is solution of the 1st order dy differential equation 3, c1 is arbitrary constant. dx As is solution of the differential equation for every value of c1, hence it is known as general solution. Examples y sin x y cos x C y 6 x e x y 3 x 2 e x C1 y x 3 e x C1x C2 Observe that the set of solutions to the above 1st order equation has 1 parameter, while the solutions to the above 2nd order equation depend on two parameters. Families of Solutions Example 9yy ' 4 x 0 Solution 9yy ' 4 x dx C 1 9 y ( x )y '( x )dx 4 xdx C1 9y 2 9ydy 2x C1 2x 2 C1 9y 2 4 x 2 2C1 2 2 C1 y 2 x2 This yields C where C . 4 9 18 Observe that given any point (x0,y0), there is a unique solution curve of the above equation which curve goes through the given point. The solution is a family of ellipses. Origin of Differential Equations 1.Geometric Origin Solution 1. For the family of straight lines y c1 x c2 the differential equation is d2y 0 2 dx 2. For the family of curves . A. y ce x2 2 y c1e c2 e 2x B. the differential equation is dy xy dx 3 x the differential equation is d 2 y dy 6y 0 2 dx dx 16