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Ordinary Differential Equations

Ordinary Differential Equations. Dr. Ahmed Elmoasry. 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,

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Ordinary Differential Equations

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  1. Ordinary Differential Equations Dr. Ahmed Elmoasry

  2. 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

  3. Partial Differential Equation Examples: 1. u is dependent variable and x and yare independent variables, and is partial differential equation. 2. 3. u is dependent variable and x and tare independent variables

  4. 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

  5. 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

  6. 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.

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

  8. Examples of linear diff Eq.

  9. Examples of non linear diff Eq. Coefficient depends on y Power not 1

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

  11. 1st – order differential equation 1. Derivative form: 2.Differential form: . 3. General form: or Differential Equation Chapter 1

  12. First Order Ordinary Differential equation Differential Equation Chapter 1

  13. Second order Ordinary Differential Equation Differential Equation Chapter 1

  14. 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

  15. Solution of Differential Equation Differential Equation Chapter 1

  16. Examples y=3x+c , is solution of the 1st order differential equation dy/dx=3, 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.

  17. Examples Differential Equation Chapter 1

  18. Implicit & Explicit solutions Differential Equation Chapter 1

  19. Families of Solutions Example Solution Differential Equation Chapter 1 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.

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