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. solution (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. A solution obtained by giving particular values to
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. In equation (2) assume an arbitrary relation
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. The Partial Differential equation of the form
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’s form
of pde , it’s solution is given by

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

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

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

17. Complete solution is given by
SOLVED PROBLEMS
1.Solve the pde and 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 pde and find the complete, general and singular solutions

19. The complete solution is given by
with

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

21. Solution 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 5.Solve the pde Solution 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. Integrating on both sides
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 pde as 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 pde as 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. auxilary equations are
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

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

45. Integrating on both sides

46. Integrating on both sides

47. HOMOGENEOUS LINEAR PDE WITH CONSTANT COEFFICIENTS
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. or Rules to find complementary function
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. Case 1 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

51. Rules to find the particular Integral
Case 2
If the equation(3) has two equal roots i.e
The complete solution of (2) is given by
Rules to find the particular Integral
Consider the equation

52. Particular Integral (P.I)
Case 1 If
then P.I=
If and is
factor of then

53. P.I
If and is
factor of
then P.I
Case 2
P.I

54. Case 4 when is any function of x and y. P.I=
Expand in ascending powers of
or and operating on term by term.
Case 4 when is any function of x and y.
P.I=

55. 1.Find the solution of pde
Here
is factor of
Where ‘c’ is replaced by after integration
Solved problems
1.Find the solution of pde
Solution
The Auxiliary equation is given by

56. 2. Solve the pde Solution The Auxiliary equation is given by By taking
Complete solution
2. Solve the pde
Solution
The Auxiliary equation is given by

57. 3. Solve the pde
Solution
the A.E is given by

58. 4. Find the solution of pde
Complete solution =
Complementary Function + Particular Integral
The A.E is given by

59. Complete solution

60. 5.Solve
Solution

62. 6.Solve
Solution

63. 7.Solve
Solution

66. 7.Solve
Solution
A.E is

69. Non Homogeneous Linear PDES
If in the equation
the polynomial expression is not homogeneous, then (1) is a non- homogeneous linear partial differential equation Ex
Complete Solution
= Complementary Function + Particular Integral
To find C.F., factorize
into factors of the form

70. If the non homogeneous equation is of the form
1.Solve
Solution

72. Assignment

73. 1.Find the differential equation of all spheres of fixed radius having centre in xy-plane.
2.Solve the pde z=ax3+by3 by eliminating the arbitrary constants.
3.Solve the pde z=f(x2-y2) by eliminating the arbitrary constants.
4.solve

74. 5 .(D²+2DD+D²)z=exp(2x+3y)
6.4r-4s+t=6log(x+2y)
7.find the general solution of differential equation (D²+D+4)z=exp(4x-y)

75. TEST NOTE:- DO ANY TWO

76. 1. (D-D-1)(D-D-2)Z=EXP(2X-Y) 2
1.(D-D-1)(D-D-2)Z=EXP(2X-Y) 2.SOLVE (D3+D2D-DD2-D3)Z=EXP(X)COS2Y 3.SOLVE (X-Y)p+(X+Y)q=2XZ