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Mathematics

Mathematics

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Mathematics

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  1. Mathematics

  2. Session Differential Equations - 3

  3. Session Objectives • Linear Differential Equations • Applications of Differential Equations • Differential Equations of Second Order • Class Exercise

  4. The standard form of a linear differential equation of first order and first degree is where P and Q are the functions of x, or constants. Linear Differential Equations

  5. Linear Differential Equations type-1 where P and Q are the functions of x, or constants.

  6. It is a linear equation of the form The solution is given by Example – 1

  7. Solution Cont.

  8. Solve the following differential equation: It is a linear differential equation of the form Example -2 Solution: The given differential equation is

  9. Solution Cont. The solution is given by

  10. It is a linear differential equation of the form Example – 3

  11. Solution Cont. The solution is given by

  12. Linear Differential Equations type – 2 where P and Q are the functions of y, or constants.

  13. Solve the following differential equation: It is a linear differential equation of the form Example - 4 Solution: The given differential equation is

  14. Solution Cont. The solution is given by

  15. Applications of Differential Equations Differential equations are used to solve problems of science and engineering.

  16. Example - 5 A population grows at the rate of 5% per year. How long does it take for the population to double? Use differential equation for it. Solution: Let the initial population be P0 and let the population after t years be P, then [Integrating both sides]

  17. Hence, the population is doubled in Solution Cont. At t = 0, P = P0

  18. The slope of the tangent at a point P(x, y) on a curve is If the curve passes through the point (3, -4), find the equation of the curve. Solution: The slope of the curve at P(x, y) is Example - 6

  19. Solution Cont. The curve passes through the point (3, –4).

  20. Differential Equations of Second Order is the required general solution of the given differential equation.

  21. Solve the differential equation: Example -7 Solution: The given differential equation is

  22. Solution Cont.

  23. Example –8 Integrating (i), we get

  24. Solution Cont. Again integrating both sides of (ii), we get

  25. Thank you