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Mathematics

Mathematics. Session. Differential Equations - 1. Session Objectives. Differential Equation Order and Degree Solution of a Differential Equation, General and Particular Solution Initial Value Problems Formation of Differential Equations Class Exercise. Differential Equation.

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Mathematics

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  1. Mathematics

  2. Session Differential Equations - 1

  3. Session Objectives • Differential Equation • Order and Degree • Solution of a Differential Equation, General and Particular Solution • Initial Value Problems • Formation of Differential Equations • Class Exercise

  4. Differential Equation An equation containing an independent variable x,dependent variable y andthe differential coefficients of the dependent variable y with respect to independent variable x, i.e.

  5. Examples

  6. Order of the Differential Equation The order of a differential equation is the order of the highest order derivative occurring in the differential equation. The order of the highest order derivative Therefore, order is 2

  7. The degree of the highest order derivative is 2. Degree of the Differential Equation The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from fractions and radicals. Therefore, degree is 2.

  8. Determine the order and degree of the differential equation: Example - 1

  9. The order of the highest derivative is 1 and its degree is 2. Solution Cont.

  10. Determine the order and degree of the differential equation: Solution: We have Here, the order of the highest order is 4 and, the degree of the highest order is 2 Example - 2

  11. A differential equation in which the dependent variable y and its differential coefficients i.e. occur only in the first degree and are not multiplied together is called a linear differential equation. Otherwise, it is a non-lineardifferential equation. Linear and Non-Linear Differential Equation

  12. is a non-linear differential equation because the dependent variable y andits derivative are multiplied together. Example - 3 is a linear differential equation of order 2 and degree 1.

  13. For example: is a solution of the differential equation Solution of a Differential Equation The solution of a differential equation is the relation between the variables, not taking the differential coefficients, satisfying the given differential equation and containing as many arbitrary constants as its order is.

  14. is the general solution of the differential equation General Solution Ifthe solution of a differential equation of nth order contains n arbitrary constants, the solution is called the general solution. is not the general solution as it contains one arbitrary constant.

  15. is a particular solution of the differential equation Particular Solution A solution obtained by giving particular values to the arbitrary constants in general solution is called particular solution.

  16. Example - 4

  17. Solution Cont.

  18. Initial Value Problems The problem in which we find the solution of the differential equation that satisfies some prescribed initial conditions, is called initial value problem.

  19. Show that is the solution of the initial value problem satisfies the differential equation Example - 5

  20. is the solution of the initial value problem. Solution Cont.

  21. Assume the family of straight lines represented by Y X O Formation of Differential Equations is a differential equation of the first order.

  22. Assume the family of curves represented by where A and B are arbitrary constants. [Differentiating (i) w.r.t. x] [Differentiating (ii) w.r.t. x] Formation of Differential Equations

  23. [Using (i)] is a differential equation of second order Formation of Differential Equations Similarly, by eliminating three arbitrary constants, a differential equation of third order is obtained. Hence, by eliminating n arbitrary constants, a differential equation of nth order is obtained.

  24. Form the differential equation of the family of curves a and c being parameters. Solution: We have [Differentiating w.r.t. x] [Differentiating w.r.t. x] Example - 6 is the required differential equation.

  25. Solution: The general equation of a circle is Example - 7 Find the differential equation of the family of all the circles, which passes through the origin and whose centre lies on the y-axis. If it passes through (0, 0), we get c = 0 This is an equation of a circle with centre (- g, - f) and passing through (0, 0).

  26. Now if centre lies on y-axis, then g = 0. Solution Cont. This represents the required family of circles.

  27. Solution Cont.

  28. Thank you

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