PARTIAL DIFFERENTIAL EQUATIONS

1. PARTIAL DIFFERENTIAL EQUATIONS

2. Formation of Partial Differential equations Partial Differential Equation can be formed either by elimination of arbitrary constants or by the elimination of arbitrary functions from a relation involving three or more variables . SOLVED PROBLEMS 1.Eliminate two arbitrary constants a and b from here R is known constant .

3. (OR) Find the differential equation of all spheres of fixed radius having their centers in x y- plane. solution Differentiating both sides with respect to x and y

4. By substituting all these values in (1) or

5. 2. Find the partial Differential Equation by eliminating arbitrary functions from SOLUTION

6. By

7. 3.Find Partial Differential Equation by eliminating two arbitrary functions from SOLUTION Differentiating both sides with respect to x and y

8. Again d . w .r. to x and yin equation (2)and(3)

10. Different Integrals of Partial Differential Equation 1. Complete Integral (solution) Let be the Partial Differential Equation. The complete integral of equation (1) is given by

11. 2. Particular solution A solution obtained by giving particular values to the arbitrary constants in a complete integral is called particular solution . 3.Singular solution The eliminant of a , b between when it exists , is called singular solution

12. 4. General solution In equation (2) assume an arbitrary relation of the form . Then (2) becomes Differentiating (2) with respect to a, The eliminant of (3) and (4) if exists, is called general solution

13. Standard types of first order equations TYPE-I The Partial Differential equation of the form has solution with TYPE-II The Partial Differential Equation of the form is called Clairaut’sform of pde , it’s solution is given by

14. TYPE-III If the pdeis given by then assume that

15. The given pdecan be written as .And also this can be integrated to get solution

16. TYPE-IV The pdeof the form can be solved by assuming Integrate the above equation to get solution

17. SOLVED PROBLEMS 1.Solve the pdeand find the complete and singular solutions Solution Complete solution is given by with

18. d.w.r.to. a and c then Which is not possible Hence there is no singular solution 2.Solve the pdeand find the complete, general and singular solutions

19. Solution The complete solution is given by with

20. no singular solution To get general solution assume that From eq (1)

21. Eliminate from (2) and (3) to get general solution 3.Solve the pde and find the complete and singular solutions Solution The pde is in Clairaut’s form

22. complete solution of (1) is d.w.r.to “a” and “b”

23. From (3)

24. is required singular solution

25. 4.Solve the pde Solution pde Complete solution of above pde is 5.Solve the pde Solution Assume that

26. From given pde

27. Integrating on both sides

28. 6. Solve the pde Solution Assume Substituting in given equation

29. Integrating on both sides 7.Solve pde (or) Solution

30. Assume that Integrating on both sides

31. 8. Solve the equation Solution integrating

32. Equations reducible to the standard forms (i)If and occur in the pdeas in Or in Case (a) Put and if ;

33. where Then reduces to Similarly reduces to

34. case(b) If or put (ii)If and occur in pdeas in Or in

35. Case(a) Put if where Given pde reduces to and

36. Case(b) if Solved Problems 1.Solve Solution

37. where

38. (1)becomes

39. 2. Solve the pde SOLUTION

40. Eq(1) becomes

41. Lagrange’s Linear Equation Def: The linear partial differenfial equation of first order is called as Lagrange’s linear Equation. This eq is of the form Where and are functions x,y and z The general solution of the partial differential equation is Where is arbitrary function of and

42. Here and are independent solutions of the auxilary equations Solved problems 1.Find the general solution of Solution auxilary equations are

43. Integrating on both sides Integrating on both sides

44. The general solution is given by 2.solve solution Auxiliary equations are given by

45. Integrating on both sides

46. Integrating on both sides

47. The general solution is given by HOMOGENEOUS LINEAR PDE WITH CONSTANT COEFFICIENTS Equations in which partial derivatives occurring are all of same order (with degree one ) and the coefficients are constants ,such equations are called homogeneous linear PDE with constant coefficient

48. Assume that then order linear homogeneous equation is given by or

49. The complete solution of equation (1) consists of two parts ,the complementary function and particular integral. The complementary function is complete solution of equation of Rules to find complementary function Consider the equation or

50. The auxiliary equation for (A.E) is given by And by giving The A.E becomes Case 1 If the equation(3) has two distinct roots The complete solution of (2) is given by