Introduction to Differential Equations

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Presentation transcript:

Introduction to Differential Equations Dr. Khawaja Zafar Elahi

Differential Equations are classified by type, order , degree , homogeneity and linearity.

ordinary differential equations Definition: A differential equation is an equation containing an unknown function and its derivatives. . . y is dependent variable and x is independent variable, and this is an ordinary differential equations

ordinary differential equations Definition: A differential equation is an equation containing an unknown function and its derivatives. Examples:. 1. 2. 3. y is dependent variable and x is independent variable, and these are ordinary differential equations

Partial Differential Equation Examples: 1. u is dependent variable and x and y are independent variables, and is partial differential equation. 2. 3. 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 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 Degree 1 1 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: 1. is linear. Example: 2. is non - linear because in 2nd term is not of degree one.

3. 4. is non – linear Example: Example: is non - linear because in 2nd term coefficient depends on y. 3. Example: 4. is non – linear is non - linear because

It is Ordinary/partial Differential equation of order… and of degree…, it is linear / non linear, with independent variable…, and dependent variable….

1st – order differential equation 1. Derivative form: 2. Differential form: . 3. General form: or Differential Equation Chapter 1 Classfication of Differential Equation

First Order Ordinary Differential equation Differential Equation Chapter 1

Second order Ordinary Differential Equation Differential Equation Chapter 1 Classfication of Differential Equation

nth – order linear differential equation 1. nth – order linear differential equation with constant coefficients. 2. nth – order linear differential equation with variable coefficients Differential Equation Chapter 1

Differential Equation Chapter 1 Solution of Differential Equation Differential Equation Chapter 1

Examples y=3x+c , is solution of the 1st order differential equation , c1 is arbitrary constant. As is solution of the differential equation for every value of c1, hence it is known as general solution. Examples 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 Differential Equation Chapter 1 Example Solution 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 Solution Geometric Origin 1. For the family of straight lines the differential equation is Differential Equation Chapter 1 2. For the family of curves the differential equation is . A. B. the differential equation is

Differential Equation Chapter 1

Mathematical Models Differential Equation Chapter 1 1. Free falling stone where s is distance or height and g is acceleration due to gravity. 2. Spring vertical displacement Differential Equation Chapter 1 Differential Equation Chapter 1 where y is displacement, m is mass and k is spring constant 3. RLC – circuit, Kirchoff ’s Second Law q is charge on capacitor, L is inductance, c is capacitance. R is resistance and E is voltage

Differential Equation Chapter 1 Differential Equation Chapter 1 Mathematical Models Newton’s Low of Cooling where is rate of cooling of the liquid, is temperature difference between the liquid ‘T’ and its surrounding Ts Differential Equation Chapter 1 2. Growth and Decay y is the quantity present at any time Differential Equation Chapter 1