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UNIT-I
FOURIER SERIES
PART- B (8 & 16 Marks)
1. Find the Fourier series of sin x in –П< x< П.
2. Obtain the Fourier series for f(x) = 1+ x+x2
in (-П, П). Deduce that
1 + 1/22
+1/32
+…….. = П2
/6.
3. Expand the function f(x) = x sin x as a Fourier series in the interval
– П ≤ x ≤ П.
4. Obtain Sine series for f(x) = x 0 ≤ x ≤ l / 2
L - x l / 2 ≤ x ≤ l.
5. Calculate the first harmonics of the Fourier of f(x) from the following data:
X 0 30 60 90 120 150 180 210 240 270 300 330
F(x) 1.8 1.1 0.3 0.16 .5 1.3 2.16 1.25 1.3 1.52 1.76 2.0
6. Expand the function f(x) = x 0 < x < 1
2 - x 1 < x < 2 as a half range sine series.
7. Find the Fourier series to represent the function f(x) = | sinx | - П < x< П.
Hence deduce the sum of the series 1/1*3 +1/ 3*5 +1/5*7+ …………
8. Express f(x) = (П – x) / 2 as a Fourier series with period 2 П to be valid in
the interval 0 to 2 П.
9. Obtain the Fourier series for the function f(x) = П x 0 ≤ x ≤ 1
П (2 - x) 1≤ x≤ 2.
10. Obtain the half range cosine series for f(x) = (x -2) 2 in the interval 0<x<2.
UNIT –II
FOURIER TRANSFORMS
PART-B (8 & 16 marks)
1.
Show that the Fourier transform of e-x2/2
is e-s2/2
2. Find the Fourier transform of f(x) = x if |x| ≤ a
0 if |x| >a
3. Find Fourier cosine transform of e-x2
4. Find the Fourier sine transform of f(x) = sinx 0 < x ≤ П
0 П ≤ x < ∞
5. Find Fourier sine transform of e-ax
/x where a>0.
6. State and prove convolution theorem of Fourier transform.
7. Find the Fourier sine transform of e-x
and hence find the Fourier sine
transform of x / (1+x2
) and Fourier cosine transform of 1/ (1+x2
).
8. Derive parasevals identity for Fourier transform.
9. With usual notation prove that ∫ f(x) g(x) dx =∫ Fc(s) Gc(s) ds.
10. Express the function f(x) = {1 for |x| ≤1
0 for |x| >1 as a Fourier integral.

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Hence evaluate ∫ (sinλx cosλx)/ λ dλ and find the value of ∫ (sinλ/λ dλ.
UNIT - III
PARTIAL DIFFERENTIAL EQUATIONS
PART-B (8 Marks)
1. Form the partial differential equation by eliminating the
Arbitrary Function from the relation f (xy+z2
, x+y+z) = 0
2. Form the partial differential equation by eliminating f and g
From Z= f(y) + φ(x+y+z)
3. Solve Z = px+qy+√ (p2+q2+16).
4. Find the complete solution and singular solution of
Z = px+qy+ p2
-q2
5. Solve the following p (1+q2
) = q(z-1).
6. Find the equation of the cone satisfying the equation xp+yq=z and
Passing through the circle x2
+y2
+z2
=4; x+y+z=2.
7. Solve the partial differential equation (x2
-yz)p+(y2
-zx)q=(z2
-xy)
8. Solve (D2
-DD’-6D’2
) Z= cos (2x+y) +ex-y
9. Solve y2
p-xyq =x (z-2y)
10. Solve the following x (y2
-z2
) p+ y (z2
-x2
) q-z(x2
-y2
) =0
UNIT - IV
APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS
PART-B (16 Marks)
1. A string is stretched and fastened to two points x=0 and x=l apart.
Motion is started by displacing the string in to the form y=k (lx-x2
)
from which it is released at time t=0. Find the displacement of any
point on the string at a distance of x from one end at time t.
2. A tightly stretched string of length l has its ends fastened at x-0 and x-l.
The mid point of the string is then taken to a height h and then
released from rest in that position. Obtain an expression for the
displacement of the string at any subsequent time.
3. A tightly stretched string with fixed end points x=0 and x=l is initially
in a position is given by y(x, 0) = y0 sin3
(Пx/l). It is released from rest to
this position. Find the displacement at any time t.
4. A tightly stretched string with fixed end points x=0 and x=l is initially
at rest in its equilibrium position. If it’s set vibrating strings giving
each point a velocity λ x (l-x) show the displacement.
5. A string is stretched between two fixed points at a distance 2l apart and
the points of the string are given initial velocities v where
v= cx / l in 0< x < l
c (2l-x)/l in l < x <2l
x being the distance from one end point. Find the displacement of the

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string at any subsequent level.
6. A rod 30cm long has its ends A and B kept at 200
and 800
respectively
state conditions prevail. The temperature at each end is then suddenly
reduced to 00
and kept so .Find the resulting temperature function
u (x, t) taking x=0 at A.
7. A rod of length l has its ends A and B kept at 0o
c and 100o
c
respectively until steady conditions prevail. If the temperature at B is
Reduced suddenly to 75o
c and at the same time the temperature at A
raised at 25o
c. Find the temperature
U(x,t) at a distance x from A and at time t.
8. A rectangular plate is bounded by the lines x=0 , y=0,x=a and y=b. Its
surfaces are insulated and the temperature along two adjacent edges
are kept at 100o
c while the temperature along the other two edges are
at 0o
c. Find the steady state temperature at any point in the plate. Also
find the steady state temperature at any point of a square plate of side
“a” if two adjacent edges are kept at 100o
c and the others at 0o
c.
9. A square plate is bounded by x=0, y=0, x=l and y=l. The edge x=0 is
maintained at 100 o
c and the other three edges are kept at 0 o
c. Find
steady state temperature at any point within the plate.
10. Find the steady state temperature distribution in a rectangular plate of
side a and b insulated on the lateral surfaces and satisfying the
following boundary conditions u(0,y)=u(a,y)=u(x,b)=0 and
u (x,0)=x(a-x).
UNIT-V
THE Z- TRANSFORMS
PART-B (16 Marks)
1. State and prove final value theorem.
2. Prove Z [1/ (n+1)] =Z log Z/ (Z-1).
3. Find Z [rn
cosnθ] and Z [rn
sinnθ].
4. Find Z [an
rn
cosnθ].
5. Given that F (z) = log (1+az-1
), for |z| > |a|.
6. State and prove convolution theorem on Z-transform.
7.
Derive the difference equation from yn=(A+Bn)(-3)n
8. Derive the difference equation from un=A2n
+Bn.
9. Using Z-transform solve Un+2-5Un+1+2Un=0.
10. Find Z-1
[(3z2
+z)/ (z3
-3z2
+4)].