Engineering Mathematics-III Important University Questions Unit-i Fourier Series Two Marks

ENGINEERING MATHEMATICS-IIIIMPORTANT UNIVERSITY QUESTIONS UNIT-I FOURIER SERIES TWO MARKS 1)Determine n b in the Fourier series expansion of ( ) ( ) x x f − · π 2 1 in π 2 0 < < x with period π 2 . (May/June 2007) 2)Define root mean square value of ( ) x f in b x a < < . (M/J’07) 3)If ( ) π π π 2 0 , 5 0 , c o s < < < < ¹ ' ¹ · x i f x i f x x f and ( ) ( ) π 2 + · x f x f for all x, find the sum of the Fourier series of ( ) x f at π · x . (Nov/Dec 2007) 4)Find the value of n a in the cosine series expansion of ( ) k x f · in the interval (0,10). (Nov/Dec 2006) 5)Find the root mean square value of the function ( ) x x f · in the interval ( 0, l ). (Nov/Dec 2006) 6)State Dirichlet’s conditions for a given function to expand in Fourier series. (Nov/Dec 2003) 7)If the Fourier series of the function ( ) 2 x x x f + · in the interval π π < < − x is ( ) ∑ ] ] ] − − + ∞ ·1 2 2 sin 2 cos 4 1 3 n n nx n nx n π , then find the value of the infinite series ..... 2 3 1 2 2 1 2 1 1 + + + (Nov/Dec 2003) 8)Find the Fourier sine series of the function ( ) 1 · x f , π < < x 0 . (Apr/May 2004) 9)If the Fourier series of the function ( ) π π π 2 0 , s i n , 0 < < < < ¹ ' ¹ · x x x x f is ( ) x x x x x f sin 2 1 ...... 7 5 6 cos 5 3 4 cos 3 1 2 cos 2 1 + ] ] ] + ⋅ + ⋅ + ⋅ + − · π π deduce that 4 2 ......... 7 5 1 5 3 1 3 1 1 − · ∞ − ⋅ + ⋅ − ⋅ π .(Apr/May 2004) 10) Does ( ) x x f tan · possess a Fourier expansion ? (Nov/Dec 2005) 11) State Parseval’s Theorem on Fourier series. (Nov/Dec 2005) 12) Find n b in the expansion of 2 x as a Fourier series in ( ) π π, − . 1 (Nov/Dec 2005) 13) If ( ) x f is an odd function defined in ( -l , l ) , what are the values of 0 a and n a ? (Nov/Dec 2005) 14) Find the constant term in the Fourier series corresponding to ( ) x x f 2 cos · expressed in the interval ( ) π π, − . (Oct/Nov 2002) 15) To which value the Half range sine series corresponding to ( ) 2 x x f · expressed in the interval (0,2) converges to 2 · x ? (Oct/Nov 2002) 16) If ( ) 2 x x x f + · is expressed as a Fourier series in the interval (-2,2) to which value this series converges at 2 · x ? (Apr/May 2003) 17) If the Fourier series corresponding to ( ) x x f · inthe interval ( ) π 2 , 0 is ( ) ∑ + + ∞ 1 0 sin cos 2 nx b nx a a n n ,without finding the values of . , , 0 n n b a a Find the value of ( ) ∑ + + ∞ 1 2 2 2 0 2 n n b a a . (Apr/May 2003) 18) Find the Half range sine series for ( ) 2 · x f in π < < x 0 . (April 2001) 19) If the cosine series for ( ) x x x f sin · for π < < x 0 is given by ( ) ∑ − − − − · ∞ 2 2 , cos 1 1 2 cos 2 1 1 sin nx n x x x n Prove that 1+2 2 ........ 7 5 1 5 3 1 3 1 1 π · , ` . | − ⋅ + ⋅ − ⋅ . (April 2001) 20) What do you mean by Harmonic analysis ? (Apr/May 2005) 21) In the Fourier expansion ( ) π π π π < < < < − ¹ ¹ ¹ ' ¹ − + · x x x x x f 0 0 , 2 1 , 2 1 in ( ) π π, − , find the value of n b ,the coefficient of sinnx (Apr/May 2004) 22) Find n a in expanding x e − as Fourier series in ( ) π π, − .(May 2006) 23) State Parseval’s identity of Fourier series. (Nov/Dec 2004) 24) If ( ) ∑ + + · ∞ 1 0 3 sin cos 2 cos nt b nt a a t n n in π 2 0 ≤ ≤ t ,find the sum of the series ( ) ∑ + + ∞ 1 2 2 2 0 4 n n b a a .(Nov 2007) 25) The Fourier series of 2 x in (0,2) and that of ( ) 2 2 + x in (-2,0) are identical or not. Give reason. (Nov 2007) 26) Define the value of the Fourier series of ( ) x f at a point of discontinuity (Dec 2008) 27) If ( ) x sinh x f · is defined in π π < < − x , write the values of Fourier coefficients 0 a and n a . (Dec 2008) 28) If ] ] ] + − + − · ... 4 x 4 sin 3 x 3 sin 2 x 2 sin 1 x sin 2 x in π < < x 0 , prove that ∑ · 6 2 2 n 1 π . (Dec 2008) 29) The functions ( ) x tan x f · , ( ) , ` . | · x 1 sin x f cannot be expanded as a Fourier series. Why ? (Dec 2008) 2 30) Expand the function ( ) 1 x f · , π < < x 0 as a series of sines. (Dec 2008) 31) Find the Fourier Cosine series of ( ) x 2 cos x f · , π < < x 0 . (Dec 2008) 32) ( ) 2 x x f · , 2 x 0 ≤ ≤ which one of the following is correct a) an even function b) an odd function c) neither even nor odd (Dec 2008) 33) Define root mean square value of a function ( ) x f over the range (a,b) (Dec 2008) 34) Define Harmonic analysis. (Dec 2008) 35) Let ( ) x f be defined in ( ) π 2 , 0 by ( ) ¹ ¹ ¹ ' ¹ − + x cos x x cos 1 x f π : : π π π 2 x x 0 < < < < and ( ) ( ) x f 2 x f · + π . Find the value of ( ) π f . (May/June 2009) 36) State the Dirichlet’s condition for the convergence of the Fourier series of ( ) x f in [ ] π 2 , 0 with period π 2 . (May/June 2009) SIX MARKS 1) Determine the Fourier series for the function ( ) 2 x x f · of period π 2 in π 2 0 < < x . (May/June 2007) 2) Find the Half range cosine series for the function ( ) ( ) x x x f − · π in π < < x 0 .Deduce that 90 ....... 3 1 2 1 1 1 4 4 4 4 π · + + + . (May/June 2007) (May/June 2009) 3) Find the complex form of Fourier series for the function ( ) x e x f − · in 1 1 < < − x . (May/June 2007) 4) Determine the Fourier series for the function ( ) ¹ ' ¹ < < < < − + + − · π π x x x x x f 0 0 , 1 , 1 Hence deduce that 4 ........ 5 1 3 1 1 π · − + − . (May/June 2007) 5) By finding the Fourier cosine series for ( ) x x f · in π < < x 0 , Show that ( ) ∑ − · ∞ ·1 4 4 1 2 1 96 n n π . (Nov/Dec 2005) 6) Find the complex form of the Fourier series of the function ( ) x e x f · when π π < < − x and ( ) ( ) x f x f · + π 2 (N/D07) 7) Find the Half range cosine series of ( ) ( ) 2 x x f − · π in the interval ( ) π , 0 .Hence find the sum of the series ∞ + + + + ....... 3 1 2 1 1 1 4 4 4 . (Nov/Dec 2006) 8) Find the Fourier series as the second harmonic to represent the function given un the following data: X : 0 1 2 3 4 5 Y : 9 18 24 28 26 20. (N/D 2006) 10) Find the Fourier series expansion of period and period for the 3 function ( ) , ` . | , ` . | ¹ ' ¹ − · l l l in in x l x x f , 2 2 , 0 Hence deduce the sum of the series ( ) ∑ − ∞ · 1 4 1 2 1 n n . (Nov/Dec 2006) 11) Obtain the Fourier series of ( ) x f of period 2l and defined as follows ( ) ¹ ' ¹ < ≤ ≤ < − · l x l l x x l x f 2 0 , 0 , .Hence deduce that 4 ........ 5 1 3 1 1 π · − + − and 8 ....... 5 1 3 1 1 1 2 2 2 2 π · + + + . (Nov/Dec 2007),(Dec 2008) 12) Determine the Fourier expansion of ( ) x x f · in the interval π π < < − x . (Apr/May 2004) 13) Find the Half range cosine series for x xsin in ( ) π , 0 .(A/M 2004) 14) Obtain the Fourier series for the function ( ) ( ) ¹ ' ¹ ≤ ≤ ≤ ≤ − · 2 1 1 0 , 2 , x x x x x f π π . (Apr/May 2004) 15) Find the Fourier series of period π 2 for the function 2 , , 0 2 1 i n i n x f and hence find the sum of the series ∞ + + + + ....... 5 1 3 1 1 1 2 2 2 . (Apr/May 2004), (Apr/May 2005) 16) Obtain the Fourier expansion for x cos 1 − in π π < < − x . (March 1996) 17) Find the Fourier series for the function ( ) ( ) ¹ ' ¹ < < < < − · 2 1 1 0 , 1 , x x in in x x x f .Deduce that 8 ....... 5 1 3 1 1 1 2 2 2 2 π · ∞ + + + + . (Nov/Dec 2005) 18) Find the Fourier series for ( ) x x f cos · in the interval ( ) π π, − . (Nov/Dec 2005) 19) Find the Fourier series for ( ) 2 x x f · in ( ) π π, − .Hence find ∞ + + + + ....... 3 1 2 1 1 1 4 4 4 . (Nov/Dec 2005) (Dec 2008) 20) Expand in Fourier series of periodicity π 2 of ( ) ( ) ¹ ' ¹ < < < < − · π π π π 2 0 , 2 , x x x x x f . 4 21) Find the Fourier series expansion of the periodic function ( ) x f of period 2l defined by ( ) ¹ ' ¹ ≤ ≤ ≤ ≤ − − + · l x x l x l x l x f 0 0 , , Deduce that ( ) 8 1 2 1 2 1 2 π · ∑ − ∞ · n n (Oct/Nov 2002) 22) Obtain the half range cosine series for ( ) ( ) 2 2 x x f − · in the interval (0,2). (Dec 2008) 23) Expand ( ) 2 x x x f + · in ( ) π π, − as a full range Fourier series and hence deduce the sum of the series ∑ ∞ ·1 n 2 n 1 (Dec 2008) 24) Expand ( ) 2 x 0 , 2 x x 2 x f < < − · as a series of cosines. (Dec 2008) 25) Give the sine series of ( ) 1 x f · in ( ) π , 0 and prove that 8 2 ... 3 , 1 2 n 1 π · ∞ ∑ (Dec 2008) 26) Find the Fourier series up to second Harmonic for the data x : 0 60 120 180 240 300 360 f(x): 1 1.4 1.9 1.7 1.5 1.2 1 (Dec 2008) 27) Find the cosine series of ( ) 2 x x f · in ( ) π , 0 (Dec 2008) 28) Find Fourier series of ( ) ¹ ' ¹ < ≤ ≤ < − · l x l l x x l x f 2 0 , 0 , (Dec08) (N/D07) 29) Find the Fourier series of ( ) 2 x x f · in ( ) π 2 , 0 and periodic with period π 2 . Hence deduce that ∑ ∞ · · 1 n 6 2 2 n 1 π (May/June 2009) 30) If a is not an integer , find the complex Fourier series of ( ) ax cos x f · in ( ) π π, − . (May/June 2009) 31) Compute the first two harmonics of the Fourier series of ( ) x f given in the following table: x : 0 3 / π 3 / 2π π 3 / 4π 3 / 5π π 2 ( ) x f : 1.0 1.4 1.9 1.7 1.5 1.2 1.0 (May/June 2009) UNIT-II FOURIER TRANSFORM TWO MARKS 1) Write the Fourier transform pair. (Nov/Dec 2007) 2) If ( ) S F C is the Fourier cosine transform of f(x) , prove that the Fourier cosine transform of f(ax) is , ` . | a s F a C 1 . (Oct 2002) 3) If F(S) is Fourier transform of f(x) , write the Fourier transform of f(x)cos(ax) in terms of F. (April 2003) 5 4) State the Convolution theorem for Fourier transforms. (M/J 2009) (N/D2005),(Apr’03) 5) If F(S) is Fourier transform of f(x) , then find the Fourier transform of f(x-a). (Nov/Dec 2003) 6) If ( ) S F s is the Fourier Sine transform of f(x), show that ( ) [ ] ( ) ( ) [ ] a s F a s F ax x f F S S S − + + · 2 1 cos . (Nov/Dec 2003) 7) Solve the integral equation ( ) λ λ − ∞ · ∫ e xdx x f cos 0 . (April 2004) 8) Find the Fourier transform of f(x) if ( ) ¹ ' ¹ · 0 1 x f : : 0 > > < a x a x . (April 2004) 9) Find the Fourier cosine transform of x e − . (Nov 2004) 10) Find a) ( ) { ¦ x f x F n and b) ( ) ¹ ' ¹ ¹ ' ¹ n n dx x f d F in terms of the Fourier transform of f(x). (Nov 2004) 11) State Fourier integral theorem. (April 2005) (May/June 2009) 12) Find the Fourier Sine transform of x 1 . (April 2005),(Dec 2008) 13) Find the Fourier transform of 0 , > − α α x e . (Nov/Dec 2005) 14) Find the Fourier cosine integral representation of ( ) ¹ ' ¹ · 0 1 x f 1 1 0 > < < x x . 15) Find the Fourier Sine transform of f(x)= x e − . (May 2006) 16) Prove that ( ) { ¦ , ` . | · a s F a ax f F 1 , a>0. (May 2006) 17) If ( ) { ¦ ( ) s f x f F · then give the value of ( ) { ¦ ax f F . (May 2006) 18) Find the Fourier transform of ( ) ¹ ' ¹ · 0 1 x f 1 1 > < x x . (May 2006) 19) Find the Fourier cosine transform of f(x) defined as ( ) ¹ ¹ ¹ ' ¹ − · 0 2 x x x f for for for 2 2 1 1 0 > < < < < x x x . (Nov/Dec 2006) 20) P.T ( ) [ ] ( ) a s F x f e F iax + · where ( ) [ ] ( ) s F x f F · . (M/J 2007) 21) Write down the Fourier cosine transform pair formulae. (M/J 2007) 22) If ( ) [ ] ( ) s f x f F · prove that ( ) [ ] ( ) s f e a x f F ias − · − . (N/D 2003),(Nov 2005) 23) Prove that if ( ) { ¦ ( ) s F x f F · , then ( ) { ¦ ( ) s F isa e a x f F · − (Dec 2008) 24) Find the Fourier transform of ( ) x f defined by ( ) ¹ ' ¹ · 0 1 x f , , otherwise b x a < < .(Dec 2008) 6 25) Find the Fourier Cosine transform of ( ) ¹ ' ¹ · 0 x x f , , π π ≥ < < x x 0 . (Dec 2008) 26) If ( ) { ¦ ( ) s F x f F · , prove that ( ) { ¦ ( ) s F 2 ds 2 d x f 2 x F − · . (Dec 2008) 27) Find the Fourier sine transform of x 1 . (Dec 2008) 28) State Parseval’s identity on complex Fourier transforms. (Dec 2008) 29) If ( ) { ¦ ( ) s F x f F · then prove that ( ) { ¦ ( ) a s F x f iax e F + · . (Dec 2008) 30) State Modulation theorem in Fourier transform. (Dec 2008) Give a function which is self reciprocal under Fourier sine and cosine transforms. (Dec 2008) 31) If ( ) { ¦ ( ) s c F x f c F · , then prove that ( ) ( ) ( ) ( ) s c F ds d x xf s F − · . (Dec 2008) SIX MARKS 1) Find the Fourier transform of f(x) = ¹ ' ¹ 0 1 otherwise x for 1 ≤ .Hence prove that ∫ ∫ ∞ ∞ · , ` . | · 0 0 2 2 sin sin π dx x x dx x x . (Nov 2002),(Nov/Dec 2003) 2) Find the Fourier transform of ( ) ¹ ' ¹ · 0 sin x x f ∞ ≤ ≤ < < x x π π 0 . (Nov 2002) 3) Find the Fourier cosine transform of 2 , 0 ) 2 ( 4 ) 1 ( 3 ) ( ≥ · − − − + n n y n y n y .Deduce that ∫ ∞ − · + 0 8 2 8 16 2 cos e dx x x π , ∫ ∞ − · + 0 8 2 8 16 2 sin e dx x x x π . 4) State and prove the Convolution theorem for Fourier transforms. (Nov 2002) 5) Find the Fourier transform of ( ) 0 , > − a e x a . Deduce that (i) ( ) ∫ ∞ · + 0 3 2 2 2 4 1 a dx a x π , (ii) { ¦ ( ) 2 2 2 2 2 s a as i xe F x a + · − π . (April 2003) 6) Find the Fourier Sine transform of 2 2 x xe − . (April 2003) 7) Find the Fourier cosine transform of 2 2 x a e − .Hence evaluate the Fourier Sine transform of 2 2 x a xe − . (Nov/Dec 2006) 8) Find the Fourier transform of 2 2 x a e − . Hence prove that 2 2 x e − is self- reciprocal. (May 2006), (May 2007) 7 9) Find the Fourier cosine transform of ( ) ¹ ' ¹ − · 0 1 2 x x f otherwise x 1 0 < < . Hence prove that ∫ ∞ · , ` . | − 0 3 . 16 3 2 cos cos sin π dx x x x x x (April 2003) 10) Derive the Parseval’s identity for Fourier transforms. (April 2003) 11) Find the Fourier Sine and cosine transform of x e − . Hence find the Fourier Sine transform of 2 1 x x + and Fourier cosine transform of 2 1 1 x + . (Nov/Dec 2003) 12) Show that Fourier transform of ( ) ¹ ' ¹ − · 0 2 2 x a x f : : a x a x > < is ] ] ] − 3 a cos a a sin 2 2 λ λ λ λ π . Hence deduce that ∫ ∞ · − 0 3 . 4 cos sin π dt t t t t (Apr/May 2004) 13) Find the Fourier Sine and cosine transform of ( ) ¹ ¹ ¹ ' ¹ − · 0 2 x x x f : : : 2 2 1 1 0 > < < < < x x x (Apr/May 2004) 14) If ( ) λ f is the Fourier transform of f(x) , find the Fourier transform of f(x- a) and f(ax). (Apr/May 2004) 15) Verify Parseval’s theorem of Fourier transform for the function ( ) ¹ ' ¹ · −x e x f 0 : : 0 0 > < x x . (Apr/May 2004) 16) Find the Fourier transform of f(x), ( ) ¹ ' ¹ − · 0 1 2 x x f : : 1 1 > ≤ x x . Hence evaluate (i) ∫ ∞ , ` . | − 0 3 2 cos cos sin dx x x x x x (ii) ∫ ∞ · , ` . | − 0 2 3 15 cos sin π ds s s s s . (April 2005) (Dec 2008) 17) Find the Fourier Sine transform of ( ) 0 > − a x e ax . (Nov/Dec 2006) 18) Find the Fourier Sine and cosine transform of x e 2 − . Hence find the value of the following integrals (i) ( ) ∫ ∞ + 0 2 2 4 x dx (ii) ( ) ∫ ∞ + 0 2 2 2 4 dx x x . (A.U.Model Qu) 19) Evaluate (i) ( )( ) ∫ ∞ + + 0 2 2 2 2 b x a x dx (ii) ( )( ) ∫ ∞ + + 0 2 2 4 1 x x dx using Fourier transform. (Nov/Dec 2008) 20) Find the Fourier Sine and cosine transform of 1 − n x . (May 2006) 21) Using Parseval’s identity for Fourier cosine transform of ax e − evaluate ( ) ∫ ∞ + 0 2 2 2 x a dx . (Nov/Dec 2007) 8 22) Find the Fourier Sine transform of ( ) 0 , > − a e ax . Hence find [ ] ax S xe F − . Hence deduce the inversion formula. (May/June 2007) 23) Find the Fourier Sine transform of f(x) defined as ( ) ¹ ' ¹ · 0 sin x x f where where a x a x > < < 0 . (Dec 2008) 24) Find the Fourier transform of ( ) ¹ ' ¹ − · 0 x 1 x f otherwise for 1 x ≤ . Hence find the values of (i) ∫ ∞ , ` . | 0 dt 4 t t sin and (ii) ∫ ∞ , ` . | 0 dx 2 x x sin (Dec 2008) 25) Find the finite sine and cosine transform of ( ) 2 x 1 x f , ` . | − · π in the interval ( ) π , 0 . (Dec 2008) 26) Find the Fourier transform of ( ) ¹ ' ¹ − · 0 x a x f , , a x a x > < . (Dec 2008) 27) Evaluate ∫ ∞ , ` . | + , ` . | + 0 2 x 2 b 2 x 2 a dx 2 x using Parseval’s identity. (Dec 2008) 28) Find the Fourier transform of ( ) x f if ( ) ¹ ' ¹ · , 0 , 1 x f if if 0 a x a x > > < . Hence deduce that ∫ ∞ · , ` . | 0 2 dt 2 t t sin π . (May/June 2009) 29) Find the Fourier Cosine transform of 2 2 x a e − for any a>0 and hence prove that 2 / x e 2 − is self-reciprocal under Fourier Cosine transform. (May/June 2009) 30) Find the Fourier transform of ( ) ¹ ¹ ¹ ' ¹ − · , 0 , 2 x 2 a x f if if 0 a x a x > > < . Hence deduce that ∫ ∞ · − 0 4 dt 3 t t cos t t sin π . (May/June 2009) 31) Find , ` . | −ax e C F , , ` . | + 2 x 1 1 C F and , ` . | + 2 x 1 x C F . (Hence C F stands for Fourier Cosine transform) (May/June 2009) UNIT – III PARTIAL DIFFERENTIAL EQUATIONS 9 TWO MARKS 1) Solve y x z sin 2 2 · ∂ ∂ . (May/June 2007) 2) Find the complete integral of pq p y q x pq z + + · . (M/J 2007) (Dec 2008) 3) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the equation ( ) ( ) ( ) α 2 2 2 2 cot z b y a x · − + − .(Nov/Dec 2007)(May/Jun 2009) 4) Find the complete solution of the PDE 0 4 2 2 · − + pq q p .(Nov/Dec2007) 5) Form the PDE of all spheres whose centers lie on the z-axis. (N/D 2006) 6) Find complete integral of the PDE ( ) ( ) z q y p x − · − + − 3 2 1 .(N/D’06) 7) obtain the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the equation ( ) ( ) 1 2 2 2 · + − + − z b y a x . (Nov/Dec 2003) 8) Find the general solution of 0 9 12 4 2 2 2 2 2 · ∂ ∂ + ∂ ∂ ∂ − ∂ ∂ y z y x z x z .(Nov/Dec 2003) 9) Eliminate the arbitrary function ‘f ’ from , ` . | · z xy f z and form the PDE. (Apr/May 2004) 10) Find the Complete integral of p+q=pq. (Apr/May 2004) 11) Find the PDE of all planes passing through the origin. (N/D 2005) 12) Find the particular integral of y x Z D D D D D D 2 s i n 1 2 4 3 3 2 2 3 + · ′ + ′ − ′ − . (Nov/Dec 2005) 13) Find the PDE of all the planes having equal intercepts on the x & y axis. (Nov/Dec 2005) 14) Find the solution of 2 2 2 z qy px · + . (Nov/Dec 2005) 15) Find the PDE of all spheres having their centers on the line x=y=z . (Oct/Nov 2002) 16) Solve 0 8 4 2 3 3 2 3 2 3 3 3 · ∂ ∂ + ∂ ∂ ∂ − ∂ ∂ ∂ − ∂ ∂ y z y x z y x z x z . (Oct/Nov 2002) 17) Solve 0 2 3 3 2 3 Z D D D D . (Apr/May 2003) 10 18) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from ( )( ) 2 2 2 2 b y a x z + + · . (Nov/Dec 2004) 19) Solve ( ) 0 1 2 · − ′ + ′ − Z D D D D (Nov 2004) 20) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from n n by ax z + · . (Apr/May 2005) (Dec 2008) 21) Solve 0 3 3 2 3 Z D D D D 22) Form the PDE by eliminating the arbitrary function from z=f(xy). 23) Write down the complete solution of 2 2 1 q p c qy px z + + + + · . 24) Form the partial differential equation by eliminating the arbitrary constants ‘a’ and ‘b’ from 3 by 3 ax z + · (Apr 1995) (Dec 2008) 25) Find the singular solution of 1 2 q 2 p qy px z + + + + · . (Dec 2008) 26) Find the general solution of z qy px · + . (Dec 2008) 27) Find the particular integral of y 4 x 3 e Z D D 4 2 D + · , ` . | ′ − .(Dec 2008) 28) Solve 1 q p · + (Dec 2008) 29) Solve 0 Z D 2 D 3 D · , ` . | ′ + + (Dec 2008) 30) Form the p.d.e by eliminating a and b from ( ) b y x a z + + · .(Dec 2008) 31) Solve y x q p + · + (Dec 2008) 32) Give the general solution of 0 y x z 2 · ∂ ∂ ∂ .(Dec 2008) 33) Solve 0 Z 2 D 2 D D 3 2 D (Dec 2008) 34) Form the partial differential equation by eliminating f from the relation y x 2 y 2 x f z + + , ` . | + · .(May/June 2009) SIX MARKS. 1) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the expression ( ) ( ) 2 2 2 2 c z b y a x · + − + − . (May/June 2007) 11 2) Solve y x Z D D D D 2 s i n 5 2 5 2 2 2 . (May/June 2007) 3) Solve ( ) ( ) ( ) 2 2 2 2 y x z q z x y p z y x − · + + + . (May/June 2007) 4) Solve p(1+q)=qz. (May/June 2007) 5) Solve zx xyq p z y x 2 2 ) ( 2 2 2 · + − − . (Nov/Dec 2007) (May/June 2009) 6) Solve y x e y x Z D D D D 4 3 2 2 3 2 2 . (Nov/Dec 2007) 7) Solve ( ) 2 2 2 2 2 y x q p z + · + . (Nov/Dec 2007) 8) Solve y x z D D D D s i n ) 4 3 ( 2 2 + · ′ − ′ + . (Nov/Dec 2007) 9) Find the singular integral of 2 2 q pq p qy px z + + + + · . (Nov/Dec 2006) 10) Solve x y Z D D D D s i n 6 5 2 2 . (Nov/Dec 2006) 11) Solve ( ) ( ) ( ) x y q z x p y z 3 2 2 4 4 3 − · − + − .(N/D 2006), (N/D 2003) 12) Solve y x e y x z D D D D − + · ′ + ′ + 2 2 2 ) 2 ( . (Nov/Dec 2006) 13) Find the singular integral of PDE 2 2 q p qy px z − + + · .(N/D 2003) 14) Solve y x e z D D D D y x 2 s i n 3 ) 5 4 ( 2 2 2 − + · ′ − ′ + − .(N/D 2003) 15) Find the general solution of ( ) ( ) ( ) 2 2 2 2 2 2 x y z q z x y p y z x − · − + − . 16) Solve 2 2 3 2 2 ) 2 ( ) 2 3 3 2 ( y x e e z D D D D D D − + · + ′ + − ′ + ′ − . (N/D 2003) 17) Solve ( ) ( ) ( ) y x z q x z y p z y x − · − + − . (Apr/May 2004) 12 18) Solve y x e y x z D D D D + 2 3 2 3 2 s i n ) 6 7 ( . (A/M 2004) 19) Solve ( ) ( ) 2 2 y x zq y x zp y x + · − + + . (Nov/Dec 2005) 20) Solve y x e z D D D D D D + · + ′ + + ′ + ′ + 2 2 2 ) 1 2 2 2 ( . (N/D 2005) 21) Solve 2 2 1 q p qy px z + + + + · . (May/June 2009) (Nov/Dec 2005),(Apr/May 2004) 22) Solve x y y z y x z x z cos 6 2 2 2 2 2 · ∂ ∂ − ∂ ∂ ∂ + ∂ ∂ . (Nov/Dec 2005) 23) Solve ( ) ( ) ( ) y x z q xz y x p yz y x + · − + + + + 2 2 2 2 .(N/D 2005) 24) Solve ( ) y x e z D D D D y x − + · ′ − ′ − + 4 s i n ) 2 0 ( 5 2 2 .(Nov/Dec 2005) 25) Solve 2 2 q p z + · . (Nov/Dec 2005) 26) Solve y x e y x z D D D D + + · ′ − ′ + 3 2 2 2 ) 6 ( . (Nov/Dec 2005) 27) Form the PDE by eliminating the arbitrary functions f and g in ( ) ( ) y x g y x f z 2 2 3 3 − + + · . (Oct/Nov 2002) 28) Solve ( ) ( ) ( ) ( ) y x y x q x yz p xz y − + · − + − . (Oct/Nov 2002) 29) Solve y e x y z D D D D x + + · ′ − ′ − 6 2 2 ) 3 0 ( . 30) Solve 2 2 2 1 q p z + + · . (April 1996) (Dec 2008) 31) Solve ( ) ( ) z x q y x p z y + · + − − 2 2 . (Apr/May 2003) 32) Form the PDE by eliminating the arbitrary functions f and g in ( ) ( ) x g y y f x z 2 2 + · 33) Form the p.d.e by eliminating the function f and g from ( ) ( ) y 2 x xg y 2 x f z + + + · (Dec 2008) 34) Solve z xq yp · + (Dec 2008) 35) Solve 2 z 2 y 2 q 2 x 2 p · + (Dec 2008) 36) Solve ( ) y 2 x 3 y 2 x sin z D 2 D 2 3 D + + · , ` . | ′ − (Dec 2008) 37) Form the p.d.e by eliminating the arbitrary function f and g from ( ) ( ) y 2 x xg y 2 x f z + + + · (Dec 2008) 13 38) Solve 0 xz xyq p 2 z 2 y · + − , ` . | + (Dec 2008) 39) Solve x cos y t 6 s r · − + (Dec 2008) 40) Obtain complete solution of the equation pq 2 qy px z − + · (Dec 2008) 41) Solve y x 2 c o s z 2 D 6 D D 2 D . (Dec 2008) 42) Solve xy yzq xzp · + (Dec 2008) 43) Solve y x 2 e z 2 D 2 D D 5 2 D 2 (Dec 2008) 44) Find the complete solution of 2 z pqxy · (May/June 2009) 45) Solve the equation y 2 x s i n y x e Z 2 D 2 D . (May/June 2009) UNIT-IV APPLICATIONS OF PDE TWO MARKS 1) Classify the following second order partial differential equations: i) 0 16 8 6 4 4 2 2 2 2 2 · − ∂ ∂ − ∂ ∂ − ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ u y u x u y u y x u x u . (Apr/May 2003) ii) 2 2 2 2 2 2 , ` . | ∂ ∂ + , ` . | ∂ ∂ · ∂ ∂ + ∂ ∂ y u x u y u x u . (Apr/May 2003) iii) 0 3 2 2 2 2 · − + + − u u u x xyu u y x yy xy xx . (Nov/Dec 2003) iv) 0 7 2 2 2 · + + + + y x yy xx u u u u y . (Nov/Dec 2003) v) 0 · + yy xx xu u . (Apr/May 2004) 2) Classify the partial differential equations t u x u ∂ ∂ · ∂ ∂ 2 2 2 1 α . (May/June 2007) 3) Classify the following PDE i) x u xu u x yy xy xx · + − + 4 ) 1 ( 2 . (March 1998), (Apr/May 2004) ii) 0 2 ) 1 ( 2 2 2 · − + + + x yy xy xx u u y xyu u x . (Dec 1998) 14 4) What is the constant 2 a in the wave equation xx tt u a u 2 · ? (N/D 2004) 5) In the diffusion equation 2 2 2 x u t u ∂ ∂ · ∂ ∂ α what does 2 α stand for? (N/D 2005) (Dec 2008) 6) What is the basic difference between the solutions of one dimensional wave equation and one dimensional heat equation? (Nov/Dec 2005) 7) What are the possible solutions of one dimensional wave equation? (May/June 2006) 8) Explain the various variables involved in one dimensional wave equation. (April 1995),(Nov 1995) 9) A tightly stretched string of length 2L is fastened at both ends. The midpoint of the string is displaced to a distance ‘b’ and released from rest in this position. Write the initial conditions. (May 2006) 10) Write the initial conditions of the wave equation if the string has an initial displacement but no initial velocity. (A.U.Tri. Nov/Dec 2008) 11) Write the boundary conditions and initial conditions for solving the vibration of string equation , if the string is subjected to initial displacement f(x) and initial velocity g(x). (Nov/Dec 2006),(April 1998) 12) State one dimensional heat equation with initial and boundary conditions. (Nov/Dec 2006) 13) In steady state conditions derive the solution of one dimensional heat flow equation. (Nov/Dec 2005) 14) An insulated rod of length 60 cm has is ends A and B maintained at C  20 and C  80 respectively. Find the steady state solution of the rod. (Nov/Dec 2003) 15) A rod 30 cm long has its ends A and B kept at C  20 and C  80 respectively until steady state conditions prevail. Find the steady state temperature in the rod. (Apr/May 2004) 16) State any two laws which are assumed to derive one dimensional heat equation. (Nov/Dec 2004) 17) State Fourier law of heat conduction. (Apr/May 2005) 18) What are the possible solutions of one dimensional heat equation? (May 2000) (May/June 2009) 15 19) How many boundary conditions are required to solve completely 2 2 2 x u t u ∂ ∂ · ∂ ∂ α (April 1995) 20) Define temperature gradient. (Nov 1995) 21) State the assumptions made in the derivation of one dimensional wave equation. (April 1995), (Nov 1995),(Nov 2007) (Dec 2008) 22) Write the steady state heat flow equation in two dimension in Cartesian & Polar form. (Nov/Dec 2005) 23) Write any two solutions of the Laplace equation obtained by the method of separation of variables. (April 2003) 24) In two dimensional heat flow, the temperature at any point is independent of which coordinate? 25) Explain the term steady state. 26) Classify the p.d.e 0 yy u xy u 2 x 2 5 xx u 2 x 4 2 x 1 · + , ` . | + + , ` . | + , ` . | + (Dec 2008) 27) State the empirical laws used in deriving one-dimensional heat flow equation. (Dec 2008) 28) Write the product solutions of 0 u r ru rr u 2 r · + + θ θ . (Dec 2008) 29) What is the equation governing the two dimensional heat flow steady state and also write its solution. (Dec 2008) 30) Classify the p.d.e y 3 x 2 e 2 y u 2 y x u 2 2 2 x u 2 + · ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ . (Dec 2008) 31) Write the various possible solutions of the Laplace equation in two dimensions. (Dec 2008) 32) A infinitely long uniform plate is bounded by the edges l x , 0 x · · and the ends right angles to them. The breadth of the edges 0 y · is l and is maintained at ( ) x f . All the other edges are kept at . C 0  Write down the boundary condition in mathematical form. (Dec 2008) 33) Write any two assumptions made while deriving the partial differential equation of transverse vibrations of a string. (Dec 2008) 34) Define steady state. Write the one dimensional heat equation in steady state. (Dec 2008) 16 35) Write all the solutions of Laplace equation in Cartesian form, using the method of separation of variables. (Dec 2008) 36) Verify that ( ) ( ) at cosh x cosh y λ λ − · is a solution of . 2 2 2 2 2 x y a t y ∂ ∂ · ∂ ∂ (M/J’09) 12 MARKS 1) A tightly stretched string of length ‘ l ’ has its ends fastened at x=0 & x=l . The midpoint 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. (Nov 2002) 2) A tightly stretched flexible string has its ends fixed at x=0 and x=l. At time t=0 , the string is given a shape defined by ), ( ) ( 2 x l kx x f − · where k is a constant , and then released from rest. Find the displacement of any point x of the string at any time t > 0. (April 2003) 3) A tightly stretched string with fixed end points x=0 and x=l is initially in a position given by , ` . | · l x y x y π 3 0 sin ) 0 , ( . It is released from rest from this position. Find the displacement at anytime ‘ t ’. (Nov 2004) 4) A tightly stretched string of length ‘ 2l ’ has its ends fastened at x=0 , x=2l. The midpoint of the string is then taken to height ‘ b ’ and then released from rest in that position. Find the lateral displacement of a point of the string at time ‘ t ’ from yhe instant of release. (May 2005) 5) A string of length ‘ l ’ has its ends x=0 , x=l fixed. The point where 3 l x · is drawn aside a small distance ‘ h ’,the displacement ) , ( t x y satisfies . 2 2 2 2 2 x y a t y ∂ ∂ · ∂ ∂ Find ) , ( t x y at any time ‘ t ’. 6) An elastic string of length ‘ 2l ’ fixed at both ends is disturbed from its equilibrium position by imparting to each point an initial velocity of magnitude ). 2 ( 2 x lx k − Find the displacement function ) , ( t x y . (May ‘06) 7) A uniform string is stretched and fastened to two points ‘ l ’ apart. Motion is started by displacing the string into the form of the curve ), ( x l kx y − · and then releasing it from this position at time t=0. Find the displacement 17 of the point of the string at a distance ‘ x ’ from one end at time ‘ t ’. (A.U.Tri. Nov/Dec 2008) (Dec 2008) (May/June 2009) 8) If a string of length ‘ l ’ is initially at rest in its equilibrium position and each of its points is given a velocity ‘ v ’ such that ¹ ' ¹ − · ) ( x l c cx v for for l x l l x < < < < 2 2 0 show that the displacement at any time‘ t ’ is given by ] ] ] + − · ... 3 sin 3 sin 3 1 sin sin 4 ) , ( 3 3 2 l at l x l at l x a c l t x y π π π π π . (Nov/Dec2008) 9) 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 ¹ ¹ ¹ ' ¹ − · ) 2 ( x l l c l cx v in in l x l l x 2 0 < < < < ‘ x ’ being the distance from one end point .Find the displacement of the string at any subsequent time. (April/May 2004) 10) The ends A and B of a rod ‘ l ’ cm long have the temperatures C  40 and C  90 until steady state prevails. The temperature at A is suddenly raised to C  90 and at the same time that at B is lowered to C  40 . Find the temperature distribution in the rod at time ‘ t ’ . Also show that the temperature at the midpoint of the rod remains unaltered for all time , regardless of the material of the rod. (April 2003) 11) A metal bar 10 cm long with insulated sides , has its ends A and B kept at C  20 and C  40 until steady state conditions prevail. The temperature at A is then suddenly raised to C  50 and at the same instant that at B is lowered to C  10 . Find the subsequent temperature at any point of the bar at any time . (Nov/Dec 2005) 18 12) The ends A and B of a rod ‘ l ’cm long have their temperatures kept at C  30 and C  80 , until steady state conditions prevail. The temperature at the end B is suddenly reduced to C  60 and that of A is increased to C  40 . Find the temperature distribution in the rod after time ‘ t ’. (M/J’ 07) 13) The boundary value problem governing the steady state temperature distribution in a flat, thin , square plate is given by , 0 2 2 2 2 · ∂ ∂ + ∂ ∂ y u x u a x < < 0 , a y < < 0 0 ) 0 , ( · x u , , ` . | · a x a x u π 3 sin 4 ) , ( , a x < < 0 0 ) , 0 ( · y u , 0 ) , ( · y a u , a y < < 0 . Find the steady-state temperature distribution in the plate. (Nov 2002) 14) A rectangular plate with insulated surface is 10 cm wide so long compared to its width that it may be considered infinite length. If the temperature along short edge y=0 is given by , ` . | · 10 sin 8 ) 0 , ( x x u π when 10 0 < < x , while the two long edges x=0 and x=10 as well as the other short edge are kept at C  0 , find the steady state temperature function ) , ( y x u . (Nov 2003) 15) An infinitely long rectangular plate with insulated surface is 10 cm wide. The two long edges and one short edge are kept at zero temperature while the other short edge x=0 is kept at temperature given by ¹ ' ¹ − · ) 10 ( 20 20 y y u for for 10 5 5 0 ≤ ≤ ≤ ≤ y y . Find the steady state temperature in the plate. (Nov/Dec 2005), (Nov 2004) (Dec 2008) 16) A rectangular plate with insulated surface is 10 cm wide and so long compared to its width that it may be considered infinite in length without introducing appreciable error. The temperature at short edge y=0 is given 19 by ¹ ' ¹ − · ) 10 ( 20 20 x x u for for 10 5 5 0 ≤ ≤ ≤ < x x and all the other three edges are kept at C  0 . Find the steady state temperature at any point in the plate. (May 2005) 17) Find the steady state temperature distribution in a rectangular plate of sides a and b insulated at the lateral surface and satisfying the boundary conditions 0 ) , ( ) , 0 ( · · y a u y u for b y ≤ ≤ 0 , 0 ) , ( · b x u and ) ( ) 0 , ( x a x x u − · for a x ≤ ≤ 0 . (Nov/Dec 2005) 18) An infinitely long plate in the form of an area is enclosed between the lines π · · y y , 0 for positive values of x. The temperature is zero along the edges π · · y y , 0 and the edge at infinity. If the edge x=0 is kept at temperature ‘ Ky(l-y)’ ’ find the steady state temperature distribution in the plate. (May 2006) 19) An infinitely long uniform plate is bounded by two parallel edges and an end at right angle to them. The breadth of this edge x=0 is π , this end is maintained at temperature as ) ( 2 y y K u − · π at all points while the other edges are at zero temperature . Find the temperature ) , ( y x u at any point of the plate in the steady state. 20) A rod of length ‘‘l ’ has its ends ‘A’ and ‘B’ kept at C 0  and C 120  respectively until steady state conditions prevail. If the temperature at ‘B’ is reduced to C 0  and kept so while that of ‘A’ is maintained, find the temperature distribution in the rod. (Dec 2008) 21) Find the steady state temperature in a circular plate of radius ‘a’ cm, which has one half of its circumference at C 0  and the other half at C 100  . (Dec 2008) 22) Find the steady state temperature distribution in a square plate bounded by the lines 20 y , 20 x , 0 y , 0 x · · · · . Its surfaces are insulated, satisfying the boundary conditions ( ) ( ) ( ) ( ) ( ) x 20 x 20 , x u & 0 0 , x u y , 20 u y , 0 u − · · · · . (Dec 2008) 23) A rectangular plate with insulated surface is 10 cm wide and so long compared to its width that it may be considered infinite in length without introducing appreciable error. If the temperature of the short edge y=0 is 20 given by x u · for 5 x 0 ≤ ≤ and ( ) x 10 − for 10 x 5 ≤ ≤ and the two long edges x=0,x=10 as well as the other short edges are kept at C  0 . Find the temperature ( ) y , x u at any point ( ) y , x of the plate in the steady state. (May/June 2009) UNIT -V Z-TRANSFORM TWO MARKS 1) Find ] ] ] ] ! n a Z n in Z-transform. (Nov/Dec 2005) 2) Find [ ] iat e Z − using Z-transform. (Nov/Dec 2005) 3) State and prove initial value theorem in Z-transform.(M/J 2006)(Dec2008) 4) Find the Z-transform of (n+1)(n+2). (May/June 2006) 5) Find the Z-transform of (n+2). (Nov/Dec 2006) 6) State the final value theorem in Z-transform. (Nov/Dec 2006) 7) Find ] ] ] n Z 1 . (May/June 2007) 8) Evaluate ] ] ] + + − 10 7 2 1 z z z Z . (May/June 2007) 9) Prove that [ ] a z z a Z n − · . (Apr/May 1999), (Apr/May 2000) 10) Prove that [ ] 2 1 ) ( a z z na Z n − · − . 11) Prove that z e k Z 1 ! 1 · ] ] ] . 12) Find [ ] c bn an Z + + 2 . 13) Find the initial and final values of the function 2 1 25 . 0 1 1 ) ( − − − + · z z z F . 14) Find the Z-transform of i) ) ( ) ( o n n f n f − · ii) ) ( ) ( o n n u n f − · iii) ) 1 ( ) ( 1 + · + n u a n f n iv) ) ( ) ( 1 n u na n f n− · . 15) What is the Z-transform of ) ( 3 1 n u n − , ` . | . 16) State the convolution property of Z-transform. (Dec 2008) 17) State and prove shifting theorem of Z-transform. 18) Find the Z-transform of 2 cos 3 π n n . 19) Find the Z-transform of ). 0 , ( ≠ b a ab n 20) Prove that [ ] [ ] ) 0 ( ) ( ) ( f z f Z T t f Z − · + . 21 21) Find the Z-transform of 1 n 1 + . (Dec 2008) 22) Prove that ( ) ( ) n a 1 n 2 a z 2 z 1 Z + · ] ] ] ] − − (Dec 2008) 23) Find the difference equation from ( ) ( ) n 2 nB A n y + · (Dec 2008) 24) State initial value theorem on Z-transform (Dec 2008) (May/June 2009) 25) Find ] ] ] ] + − 9 2 z z 1 Z . (Dec 2008) 26) Define unit impulse sequence and find its Z-transform. (Dec 2008) 27) Define convolution of two sequences. (Dec 2008) 28) Find the inverse Z-transform of 4 2 z 2 z + (Dec 2008) 29) From the difference equation of 1 y , n 2 n y 1 n y 0 · · − + , find n y in terms of z. (Dec 2008) 30) Find ( ) ( ) n f Z ,where ( ) n n f · for n= 0, 1, 2, …. (May/June 2009) SIX MARKS 1) Find ] ] ] ] − − − ) 2 ( ) 1 ( 2 3 1 z z z Z using partial fraction. (N/D2005)(Dec2008) 2) Solve the difference equation 0 ) ( 4 ) 1 ( 4 ) 2 ( · + + − + k y k y k y where . 0 ) 1 ( , 1 ) 0 ( · · y y (Nov/Dec 2005) 3) Prove that , ` . | − · ] ] ] + 1 log 1 1 z z z n Z . (Nov/Dec 2005) 4) State and prove second shifting theorem in Z-transform. (Nov/Dec 2005) 5) Using convolution theorem evaluate inverse Z-transform of ] ] ] ] − − ) 3 )( 1 ( 2 z z z Z . (Dec 2008) (May/June 2006) 6) Using Z-transform solve 2 , 0 ) 2 ( 4 ) 1 ( 3 ) ( ≥ · − − − + n n y n y n y given that . 2 ) 1 ( , 3 ) 0 ( − · · y y (May/June 2006) 7) Find ] ] ] ] − + + − − 2 2 1 ) 1 )( 1 ( ) 2 ( z z z z z Z by using method of partial fraction.(N/D 2006) 8) Find Z-transform of ) 2 )( 1 ( 1 + + n n . (Nov/Dec 2006) 9) Using convolution theorem evaluate ] ] ] ] − − − ) 2 )( 1 ( 2 1 z z z Z .(Nov/Dec 2006) 10) Find Z-transform of n a and θ n a n cos . (May/June 2007) 11) Using the Z-transform method solve 2 2 · + + n n y y given that 0 1 0 · · y y . (May/June 2007) 12) State and prove final value theorem in Z-transform.(May/Jun 2007) 13) Find the inverse Z-transform of 3 ) 1 ( ) 1 ( − + z z z . (May/June 2007) 14) State and prove first shifting theorem on Z-transform. Also find [ ] t e Z at − . 22 15) Use Z-transform to solve n n n n y y y 2 12 7 1 2 · + − + + given 0 1 0 · · y y . (Dec 2008) 16) Find [ ] iat e Z − and hence deduce the values of [ ] at Z cos and [ ] at Z sin . 17) Find ] ] ] ] − + − ) 1 ( ) 1 ( 2 1 z z z Z . 18) Prove that [ ] [ ] 1 − − · p p n Z dz d z n Z where p is any positive integer. Deduce that [ ] 2 ) 1 ( − · z z n Z and [ ] 3 2 2 ) 1 ( 2 − + · z z n Z . 19) Find the inverse Z-transform of ) 4 )( 2 ( 3 2 2 − + + z z z z . 20) Find the inverse Z-transform of ( )( ) 2 n 1 n + + . (Dec 2008) 21) Using Z-transforms, solve ( ) ( ) ( ) , 1 n , 0 n y 4 1 n y 3 2 n y ≥ · − + + + given that ( ) 3 0 y · and ( ) 2 1 y − · . (Dec 2008) 22) Find the Z-transform of the sequence 1 n 1 n f + · (Dec 2008) 23) Find the inverse Z-transform of ( ) ( ) 2 2 z z 4 2 z 2 Z F − + · using residue theorem. (Dec 2008) 24) By using convolution theorem, prove the inverse of ( )( ) b z a z 2 z + + is ( ) { ¦ 1 n a 1 n b a b n 1 + − + − − . (May/June 2009). 25) By the method of Z – transform solve ( ) ( ) ( ) n 2 n y 9 1 n y 6 2 n y · + + + + given that ( ) 0 0 y · and ( ) 0 1 y · . (May/June 2009) 26) Find the Z – transform of θ n cos and hence find ( ) θ n cos n Z . (May/June 2009) 27) Solve the equation (using Z – transform) ( ) ( ) ( ) 36 n y 6 1 n y 5 2 n y · + + − + given that ( ) ( ) . 0 1 y 0 y · · (May/June 2009) 23