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