1. Mathematics

2. Session Differential Equations - 2

3. Session Objectives  Method of Solution: Separation of Variables  Differential Equation of first Order and first Degree  Equations Reducible to Variable Separable Form  Homogeneous Differential Equations  Method of Solution  Class Exercise

4. Differential Equation of first Order and first Degree where f (x, y) is the function of x and y. A differential equation of the first order and first degree contains independent variable x, dependent variable y and its derivative

5. Separation of Variables Differential equation of the form [where C is an arbitrary constant]

6. Separation of Variables [where C is an arbitrary constant] Differential equation of the form

7. Separation of Variables [where C is an arbitrary constant] Differential equation of the form

8. Example - 1

9. Solution Cont. Integrating both sides, we get

10. Example - 2

11. Solution Cont. Integrating both sides, we get

12. Example – 3

13. Solution Cont. Integrating both sides, we get

14. Solution Cont.

15. Example - 4 Solution: We have Solve the differential equation cosy dy + cosx siny dx = 0; given that [Integrating both sides]

16. Solution Cont.

17. Reducible to Variable Separable Form Substitute ax + by + c = v to reducing variable separable form. Differential equation of the form

18. Example - 5 Solve the differential equation:

19. Solution Cont.

20. Homogeneous Function A function f (x, y) in x and y is called a homogenous function, if the degrees of each term are equal. Examples: is a homogenous function of degree 2 is a homogenous function of degree 3

21. Homogenous Differential Equations where f (x, y) and g(x, y) is a homogenous functions of the same degree in x and y, then it is called homogenous differential equation. is a homogenous differential equation as and both are homogenous functions of degree 3. Example:

22. Method of Solution (2) Substitute and in the equation. (3) The equation reduces to the form (4) Separate the variables of v and x. (5) Integrate both sides to obtain the solution in terms of v and x. (6) Replace v by to get the solution (1) Write the differential equation in the form

23. Example – 6 It is a homogeneous differential equation of degree 1.

24. Solution Cont. Integrating both sides, we get

25. Example - 7 Solve the differential equation: It is a homogeneous differential equation of degree 2.

26. Solution [Integrating both sides]

27. Solution Cont.

28. Thank you