Partial Differential Equations I Question Paper
Exam Name: Partial Differential Equations I
Category: Kenyatta University question papers
Exam Year:2008
KENYATTA UNIVERSITY
UNIVERSITY EXAMINATIONS 2007/2008
INSTITUTE OF OPEN LEARNING
SUPPLEMENTARY/SPECIAL EXAMINATION FOR THE DEGREE OF
BACHELOR OF EDUCATION AND BACHELOR OF SCIENCE
SMA 432: PARTIAL DIFFERENTIAL EQUATIONS I
DATE: _Friday 10th October, 2008_______________TIME: 1.00pm3.00pm
INSTRUCTIONS: Answer question ONE and any other TWO
Question One (30 marks)
a)
Form a partial differential equation by eliminating the arbitrary function
f from the equation.
2
2
2
x + y + z = f (x + y + z )
(6marks)
b)
Verify that the two sets of parametric Equations.
x = a sin u cosv, y = a sin u sin v, z = a cosu and
1 v2
1 v2
2av
x =
cosu, y = a
sin u, z =
1+ v2
1+ v2
2
1+ v
represent
the
same
surface
(6marks)
c)
Find the integral curves of the equations
dx
dy
dz

=
(6marks)
x + y
x + y
 (x + y + 2z)
d)
Find the orthogonal trajectories on the cone yz + zx + xy = 0 of the conics
in which it is cut by the system of planes x  y = c, where c is a
parameter.
1
(6marks)
e) Verify that the equation
( 2
y 2
+ z )dx + xydy + xzdz = 0 is integrable and determine it solution
(6marks)
Question Two (20 marks)
a)
Eliminate the constants a and b from the following equation
2z = (ax + y 2
) + b
(5marks)
b)
Find the general integral of the linear partial differential equation
z
2
z(xp  yq) = y  x
(6marks)
c)
Find the integral surface of the equation
2
2
2
2 z
(x  y) y p + ( y  x)x q = (x + y )
Through the curve
3
xz = a ,y = 0
(9marks)
Question Three (20 Marks)
Find the characteristics of equation
z =
1
2
/2 (
2
p + q ) + ( p  x)(q  y)
and determine the integral surface which passes through the xaxis. (20marks)
Question Four (20marks)
a)
Show that for the equations
f (x, y, z, p, q) = 0 and g(x, y, z, p, q) = 0 to be compatible, the condition.
?( f , g)
?( f g) ?( f , g)
?( f g)
+ p
+
+q
=
0
(10
marks)
?(x, p)
?(z, p) ?(y,q)
?(z,q)
b)
Show that the equations
xp = yq, z(xp + yq) = zxy
are compatible and solve them
(10marks)
2
Question Five (20 marks)
a)
Find the complete integral of the equation
2
p
x + q
2 y = z.
(13 marks)
b)
Find the envelope of the one parameter system of planes whose equation is
2
3
3a x  3ay + z = a
(7marks)
3
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