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# Anna University Coimbatore 2009 B.E 2 ks with Answer for Transforms and Partial Differential Equations ( Unit-3 PDE) - Question Paper

Wednesday, 16 January 2013 06:55Web

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

ANNA UNIVERSITY (2 Marks) QUESTIONS WITH ANSWERS

UNIT III

PARTIAL DIFFERENTIAL EQUATIONS

TWO MARKS

1)     Solve .

Solution:

2)     Find the complete integral of .

Solution:

Given,

This is of the form

Hence the Complete integral is

3)     Form the PDE by eliminating the arbitrary constants a and b from the equation .

Solution:

4)     Find the complete solution of the PDE .

Solution:

Given,

The complete integral is

Hence

Therefore,The complete solution is

5)     Form the PDE of all spheres whose centers lie on the z-axis.

Solution:

The equation of such spheres with centre (0,0,c) and radius r is

where r, c are constants.

Differentiating p. w.r.to x and y , we get

Substituting (3) in (2), we get

6)     Find complete integral of the PDE .

Solution:

Given,

This is of the form .

The Auxillary equation is

Now Consider

Integrating this we get

Now Consider

Integrating this we get

Hence, The General Solution is

7)     Obtain the PDE by eliminating the arbitrary constants a and b from the equation .

Solution:

,

Similarly,

8)     Find the general solution of .

Solution:

The Auxillary equation is

9)     Eliminate the arbitrary function f from and form the PDE.

Solution:

Differentiating the given equation p.w.r.to and ,

Divide by .

10)  Find the Complete integral of .

Solution:

Given

This is of the form

Solution is subject to

. This is the C.I. of

11)  Find the PDE of all planes passing through the origin.

Solution:

The general equation to a plane is

where a, b, c, d are constants.

Since passes through

Substitute the values of a & b in

.

12)  Find the particular integral of .

Solution:

13)  Find the solution of .

Solution:

Given,

This is of the form .

The Auxillary equation is

Now Consider

Integrating this we get

Now Consider

Integrating this we get

Hence, The General Solution is

14)  Solve .

Solution:

satisfies this equation , also satisfies this equation and

The roots are

Solution is where are arbitrary functions of .

15)  Solve .

Solution:

Solution is where are arbitrary functions of .

16)  Form the PDE by eliminating the arbitrary constants a and b from . Solution:

Differentiating the given equation p.w.r.to and ,

Substituting & in ,

.

17)  Solve

Solution:

The given equation can be written as

The complete solution is .

18)  Solve

Solution:

.

Solution is .

19)  Form the PDE by eliminating the arbitrary function from .

Solution:

Differentiating p.w.r.to in turn,

Equating & we get

20)  Write down the complete solution of .

Solution:

The given PDE is Clairauts Form

The complete solution is got by putting in the given PDE.

The C.I. is where and are arbitrary constants.

21)  Find the singular solution of .

Solution:

This is Clairauts Form. Hence the complete integral is

Differentiating partially w.r.to and respectively we get

and

Substituting in we get

.

22)  Find the general solution of .

Solution:

Given,

This is of the form .

The Auxillary equation is

Now Consider

Integrating this we get

Now Consider

Integrating this we get

Hence, The General Solution is .

23)  Find the particular integral of .

Solution:

Hence

24)  Solve

Solution:

The given equation is of the

The complete integral is where

Now

The complete integral is .

.

25)  Form the p.d.e by eliminating a and b from .

Solution:

Given

Differentiating the above equation p.w.r.to and ,

From & we get,

26)  Solve .

Solution:

Given, . This is of the form

Let

Substitute these values in

.

27)  Give the general solution of .

Solution:

Given,

Integrating with respect to , we get

Again Integrating with respect to ,

. Hereandare arbitrary functions of and respectively.

28)  Solve .

Solution:

Here

The solution is

29)  Form the partial differential equation by eliminating from the relation .

Solution:

Given,

Differentiating p.w.r.to we get

Differentiating p.w.r.to we get

From & we get, . .

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