1. 1Chapter 2

2. 2

3. Example 3Chapter 2

4. 4

5. EXAMPLE 5Chapter 2

6. Solution 6Chapter 2

7. 7

8. 8

9. 9 Method for solving First Order Differential Equations Differential Equations Method for solving First Order Differential Equations Differential Equations

10. Methods Variable Separable Reducible to variable separable Exact Differential Equation Integrating Factor

11. Separable Variable x is independent variable and y is dependent variable or are separable forms of the differential equation or General solution can be solved by directly integrating both the sides + c Where c is constant of integration 11 Chapter 2 DO YOU REMEMBER INTEGRATION FORMULA

12. Separation of Variables Definition A differential equation of the type y’ = f(x)g(y) is separable. Example Separable differential equations can often be solved with direct integration. This may lead to an equation which defines the solution implicitly rather than directly.

13. EXAMPLE: 13Chapter 2

14. EXAMPLE: 14Chapter 2

15. To find the particular solution, we apply the given initial condition, when x =1, y = 3 is solution of initial value problem 15Chapter 2

16. 16Chapter 2

17. 17Chapter 2

18. 18 Note1: If we have Integrating by parts Note.2. If we have Integrating by parts Note.3. If we have

19. Chapter 219

20. Chapter 220

21. Chapter 221

22. Method Homogeneous Equations Reducible to separable

23. Chapter 223 Homogenous Differential Equations A differential equation Homogenous differential equation if every t, where t  R

24. Chapter 224 Example:1. Show that differential equation is homogenous differential equation. Solution:  Differential equation is homogeneous Differential equation is homogeneous

25. Chapter 225 METHOD for solving Homogenous differential equations Substitute OR

26. Chapter 226 Using substitution the homogeneous differential equation is reduce to separable variable form. Example:2Solve the homogenous differential equation Solution: Rewriting in the form :. substitute and

27. Chapter 227 is variable separable form is general solution.

28. Chapter 228 Note.Selection of substitution Differential Equation depends on number of terms of coefficients 1.If, then take 2.If, then take 3.If, then take x = vy or y = ux

29. Chapter 229 Example:.Solve the Differential Equation by using appropriate substitution Solution: Differential equation is homogeneous as degree of each term is same, hence we can use either y = ux or x = vy as substitution Substituting y and dy in the given equation (1 / 2)

30. Chapter 230 is Separable form Integrating both the sides is general solution of the differential equation Separating variable u and x (2 / 2)

31. Chapter 231 Example: Show that differential equation is homogeneous Solution: (1 / 2)

32. Chapter 232 Let is general solution of the differential equation is Separable form Integrating both the sides (2 / 2)

33. Chapter 233 Homogeneous Differential Equation Chapter 2

34. Chapter 234 (1 / 3) Homogeneous Differential Equation Chapter 2

35. Chapter 235 (2 / 3) Homogeneous Differential Equation Chapter 2

36. Chapter 236 (3 / 3) Homogeneous Differential Equation Chapter 2

37. Chapter 237 Homogeneous Differential Equation Chapter 2

38. Chapter 238 (1 / 2) Homogeneous Differential Equation Chapter 2

39. Chapter 239 is general solution of differential equation (2 / 2) Homogeneous Differential Equation Chapter 2

40. Chapter 240 Differential Equation Chapter 2

41. Chapter 241 Differential Equation Chapter 2

42. Chapter 242 is general solution of differential equation Differential Equation Chapter 2