Exact & non differential equation

exact & non exact differential equations, integrating factor

1. EXACT & non exact DIFFERENTIAL EQUATION 8/2/2015 Differential Equation 1

2. EXACT DIFFERENTIAL EQUATION

3. EXACT DIFFERENTIAL EQUATION A differential equation of the form M ( x, y ) dx + N ( x , y ) dy = 0 is called an exact differential equation if and only if 8/2/2015 Differential Equation 3

4. SOLUTION OF EXACT D.E. The solution is given by :   8/2/2015 Differential Equation 4

5. Example : 1 Find the solution of differential equation . Solution : Let M(x, y)= and N(x , y)= Now, = , =   8/2/2015 Differential Equation 5

6. Example : 1 (cont.) The given differential equation is exact ,   8/2/2015 Differential Equation 6        

7. NON EXACT DIFFERENTIAL EQUATION

8. NON EXACT DIFFERENTIAL EQUATION For the differential equation IF then, If the given differential equation is not exact then make that equation exact by finding INTEGRATING FACTOR .   8/2/2015 Differential Equation 8

9. INTEGRATING FACTOR In general, for differential equation M ( x, y ) dx + N ( x , y ) dy = is not exact. In such situation, we find a function such that by multiplying to the equation, it becomes an exact equation. So, M(x, y)dx + N(x, y) dy = 0 becomes exact equation Here the function is then called an Integrating Factor   8/2/2015 Differential Equation 9

10. Methods to find an INTEGRATING FACTOR (I.F.) for given non exact equation: M(x , y)dx + N(x, y) dy = CASES: CASE I CASE II CASE III CASE IV

11. CASE I : If ( i.e. function of x only Then I.F. =   8/2/2015 Differential Equation 11

12. Example : 2 Solve : Solution : Let M(x, y)= and N(x , y)= Now, = , = -   8/2/2015 Differential Equation 12

13. ( Now, I.F. = = = = =   8/2/2015 Differential Equation 13 Example : 2 (cont.)

14. Multiply both side by I.F. (i.e. ), we get   8/2/2015 Differential Equation 14 Example : 2 (cont.)

15. Example : 2 (cont.) Let M(x, y)= and N(x , y)= Now, = , =   8/2/2015 Differential Equation 15

16. Example : 2 (cont.) , w hich is exact differential equation. It’s solution is :   8/2/2015 Differential Equation 16  

17. CASE II : If ) is a function of y only , say g(y), then is an I.F.(Integrating Factor).   17 8/2/2015 Differential Equation

18. Example : 3 Solve =0 Solution: Here M= and so +2 N= and so Thus, and so the given differential equation is non exact.   18 8/2/2015 Differential Equation

19. Example : 3 (cont .) Now, = - which is a function of y only . Therefore I.F.= =   19 8/2/2015 Differential Equation

20. Example : 3 (cont.) Multiplying the given differential equation by ,we have ----------------(i) Now here, M= and so N= and so Thus, and hence (i) is an exact differential equatio   20 8/2/2015 Differential Equation

21. Example : 3 (cont.) Therefore , General Solution is =c w here c is an arbitrary constant.   21 8/2/2015 Differential Equation

22. CASE III : If the given differential equation is homogeneous with then is an I.F.   22 8/2/2015 Differential Equation

23. Example : 4 Solve Solution: Here M= and so N= and so The given differential equation is non exact.   23 8/2/2015 Differential Equation

24. Example : 4 (cont.) The given differential equation is homogeneous function of same degree=3. [ = = = =   24 8/2/2015 Differential Equation

25. Example : 4 (cont.) Now, = = Thus, I.F.= Now, multiplying given differential equation by we have   25 8/2/2015 Differential Equation

26. Example : 4 (cont.) Here, M= and so N= and so Thus, and hence (i) is an exact differential equation .   26 8/2/2015 Differential Equation

27. Example : 4 (cont.) Therefore , General Solution is w here c is an arbitrary constant.   27 8/2/2015 Differential Equation

28. CASE IV : If the given differential equation is of the form is an I.F.   28 8/2/2015 Differential Equation

29. Example : 4 (cont.) Solve ( Solution: Here, M=( and so N=( and so The given differential equation is non exact .   29 8/2/2015 Differential Equation

30. Example : 4 (cont.) Now, = = So, I.F.= Multiplying the given equation by , we have   30 8/2/2015 Differential Equation

31. Example : 4 (cont.) ----------(i) Here, M= and so N= and so Thus, and hence (i) is an exact differential equation .   31 8/2/2015 Differential Equation

32. Example : 4 (cont.) Therefore , General Solution is where c is an arbitrary constant.   32 8/2/2015 Differential Equation

33. 8/2/2015 Differential Equation 33