Partial Differential Equation I Question Paper

Partial Differential Equation I

Course:Bachelor Of Science

Institution: Kenyatta University question papers

Exam Year:2009

KENYATTA UNIVERSITY

UNIVERSITY EXAMINATIONS 2009/2010
EXAMINATION FOR THE DEGREE OF BACHELOR OF ATRS , BACHELOR
OF EDUCATION AND BACHELOR OF SCIENCE

SMA 432: PARTIAL DIFFERENTIAL EQUATION I

DATE: Monday 28th December, 2009

TIME: 11.00 a.m – 1.00 p.m

INSTRUCTIONS: Answer Question ONE and any other TWO questions.

Question One

a)
Determine the integral curves.
dx
dy
dz

=
=

[4 marks]
2
xz - y
yz - x
1- z

b)
Show that the partial differential equation ap + bq + cz = 0 has a general solution of the
cy
form
Z= e
f(ay - bx) for a ? ,
0
b ? ,
0
hence solve 2 p + 3q + 5z = 0
b

[5 marks]
c)
Find the partial differential equation arising from the equation
? z
?

f (
2
2
z
2
2
+
-
by eliminating f
, x - y

[4 marks]
x y , x
y )
?
?
?
?
? x + y
?

d)
Verify whether the following Pfaffian differential equation is integrable.

(x - 3y - z)dx + (2y - 3x)dy + (z - x)dz = 0

[3 marks]

e)
Find the equation of the surface satisfying 4yz + q + 2y = 0
passing through 2
2
y + z = ,
1
x + z = 2

[5marks]

f)
Find the complete integral of the non- linear partial differential equation

2
2
xz - px - 2q xy + pq = 0 Using Charpit’s method.

[5 marks]

g)
Determine a complete solution and the singular solution of

P2 + q2 = 4z

[4 marks]
Question Two

(a)
i)
Differentiate between the terms Semi-linear and Quasi-linear as used in reference

to the Lagrange’s equation Pp + Qq = R

[2 marks]

ii)
Prove that the general solution of linear partial differential equation Pp + Qq = R

is ?(u,v) = 0 where f is an arbitrary function and u = u (x, y, z) = c, and

v = v(x, y, z) = C are two solutions of the simultaneous equation (Lagrange’s
2
dx
dy
dz
system
)
=
=

[8 marks]
p
Q
R
b)
Solve, the differential equation

xz(
2
xy + z )dx - zy( 2
z + xy)
4
dy = x

[5 marks]
c)
Find the integral curves of the equation
dx
dy
dz

=
=

[5 marks]
Cos (x + y) sin(x + y)
z
Question Three
a)
Find the equations of a system of curves on the cylinder
2
2y = x or thogonal to its
intersections with the hyperboloids of the one -parameter system xy = z + c .

[8 marks]
b)
Find the equation of the integral surface of the differential equation
?z
?z
2y(z - )
3
+ (2x - z)
= y(2x - )
3 which passes through the circles z=0, 2
2
x + y =2 x .
?x
?y

[7 marks]

Page 2 of 3

c)
Find the equation of the system of surfaces which cut orthogonally the cones of the
system. x2 + y2 + z2 = cxy .

[5 marks]

Question Four
a) Find
f (y) such that the total differential equation
yz + z

dx - zdy + f (y)dz = 0 is integrable. Hence solve it.

[8 marks]
x
b)
Solve the differential equation

zydx + ( 2
x y - zx)dy + ( 2
x z - xy)dz = 0

[6 marks]
c)
i)
Verify whether the differential equation
?
2
2
y
?

xdx + ydy + C 1 x
-
-
dz =
2
2
0
?
?
?
?
?
a
b ?
ii)
Eliminate the arbitrary constants a and b from (x - a)2 + (y - b)2
2
+ z = 4

[3marks]
Question Five
a)
i)
State the compatibility condition for the first order differential equations

f (x, y, z, p,q) = 0
and
g(x, y, z, p,q) = 0 .
[2
marks]
ii)
Show that the equations
xp - yq = x and x2 p + q = xz find their solution.

[8marks]
b)
i)
Solve the non – linear partial differential equations below using an appropriate
method
P2q(x2 + y2 ) = P2 + q .

[4 marks]

ii)
Using the transformations Y = ny
1 , Z = nz
1
reduce the equation

4
2
2
x P - yzp - z = 0 to a standard form then solve it.

[6 marks]

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