f ax ; a 0 is a periodic function b is a periodic function of x of p b. f which is assumed to have the period 2 π, where

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1 (a) (b) If () Year - Tutorial: Toic: Fourier series Time: Two hours π π n Find the fundamental eriod of (i) cos (ii) cos k k f a ; a is a eriodic function b is a eriodic function of of b. f is a eriodic function of of eriod show that ( ) of of eriod / a and f ( b) ; Verif this result for f ( ) cos ; a b. Find the Fourier series of the function ( ) f ( ) is as follows f which is assumed to have the eriod π where f ( ) π < < π f ( ) < < π () f ( ) + π < < π Find the Fourier series of the function f ( ) which is assumed to have the eriod π f ( ) is as follows f ( ) if if π < < < < π Find the Fourier series of the eriodic function ( ) if < < () f ( ) f ( ) () ( ) if < < 5 Find the Fourier series of the eriodic function ( ) f ( ) () f ( ) ; < < Find the Fourier series of the eriodic function ( ) where if π < < π if π < < π f of eriod l. 5 f < < f of eriod l. 5 ( ) + ( ) < < ( ) < < ( ) f of eriod l 5 if < < f ( ) () f ( ) π sin π if < < if < < 7 Show that each term in 5 ( ) nπ nπ f a + an cos + bn sin has the eriod l. n l l 8 If is a eriod of f ( ) show that n n... is a eriod of f ( ) Find the Fourier series of the eriodic function that is obtained b assing the voltage 5 v t v cos(π t through a half wave rectifier. ( ) ) Are the following functions odd even or neither odd nor even? 5 () () cosn () () e (5) ln. Advanced Engineering Mathematics b Erwin Kreszig..

2 Year - Tutorial: Toic: Fourier series Are the following function which are assumed to be eriodic of eriod even odd or neither even nor odd? () < < ) () 5 < < < < ) Are the following function which are assumed to be eriodic of eriod even odd or neither even nor odd? < < () < < () f ( ) π < < π () < < < < State whether the given eriodic function is even or odd. Find its Fourier series < < < < < < < < < < State whether the given eriodic function is even or odd. Find its Fourier series. < < < < < < < < 57 5 Show that < < Find the Fourier cosine series ( Half-Range Eansions) < < 7 Find the Fourier sine series ( Half-Range Eansions) < < < < < < 8 Find the Fourier cosine series ( Half-Range Eansions) of < < 57 9 Find the Fourier sine series ( Half-Range Eansions) of < < 57 Find the Fourier series of following eriodic function < < < < < < < < Advanced Engineering Mathematics b Erwin Kreszig..

3 Year - Tutorial: Toic: Fourier Integrals Show that the given integrals reresent the indicated function. < + + > Show that the given integrals reresent the indicated function. < > Show that the given integrals reresent the indicated function. < < > Show that the given integrals reresent the indicated function. + > 5 Reresent the following function as Fourier cosine integral. < < > Reresent the following function as Fourier cosine integral. < < > 7 Reresent the following function as Fourier cosine integral. > + 8 Reresent the following function as Fourier sine integral. < < > 9 Reresent the following function as Fourier sine integral. < < > Reresent the following function as Fourier sine integral. < < > Advanced Engineering Mathematics b Erwin Kreszig..

4 Year - Tutorial: Toic: Direction field and eact differential equations Plot Direction field of following. In it lot some aroimate solution curves b hand. Then solve the differential equation and comare to get an imression of the accurac of the direction field method. () () cosπ Plot Direction field of (). In it lot the articular solution satisfing the initial condition Show that following Equations are eact and solve them (- 5) d + d θ e ( dr + rdθ ) 5 ( cot + ) d cosec d Are the following roblems eact? Also solve the following initial value roblems.( & 7) d + d () d + d () 7 ( ) Show that the given function is an integrating factor and solve: 8 9 sin d + cos d e + ) d + ( b + ) d a b ( a Find an integrating factor and solve: d + d (cos + ) d sin d tan d + sec d Pag e. Advanced Engineering Mathematics b Erwin Kreszig..

5 Year - Tutorial: 5 Toic: Linear and Bernoulli s Differential Equations Find general solutions of the following differential equations.(-) k + k e 8 + cos 8 + e tan 9 cos + sin e 9 Solve the following initial value roblems.(5 & ) 5 + () 9 ( ) tanh () 9 Reduce the following equations in to Linear form and solve them.(7-) (e ( ) ) ( ) e 9 A tank contains gal of water in which lb of salt are dissolved. Fift gallons of brine each.t containing ( + e cos t) lb of dissolved salt run into the tank er minute. The miture ket uniform b stirring runs out at same rate. Find the amount of salt (t) in the tank at an time t. 9. Advanced Engineering Mathematics b Erwin Kreszig..

6 Year - Tutorial: Toic: Modeling Find orthogonal trajectories of ce 5 Find orthogonal trajectories of c 5 Find orthogonal trajectories of c 5 c 5 Find orthogonal trajectories of di 7 Can the limit of the equation L + RI E( t) as t be seen directl from the differential 5 dt equation without actuall solving it? L E t Show that T L is the time at which the current ( ) ( TL 7 I t e ) reaches about % of its R R final value. 7 E t At what time will the current I ( t) ( e TL 7 ) reach half of its theoretical maimum value? R di What L should we choose in L RI E( t) 8 dt + with 7 E E cons tan t and R ohms if we want the current to grow from to 5% of its final value within sec? 9 di Solve L RI E( t) dt + with ( ) 7 t E t e when (a) R L (b) R L dq 7 Solve R + Q E( t) with E( t ) assuming that Q() Q dt C. Advanced Engineering Mathematics b Erwin Kreszig..

7 Year - Tutorial: 7 Toic: Higher order linear differential equations Solve the following: () () ; + 9 ; Solve the given initial value roblem: () + 9 ; () ; () () ; () (). 5 () ( ) () + + ; () ; () Find a general solution: () + () () Solve the initial value roblem: () + ; () ; () () 8 ; (). (). 5 () ; () ; (). Find a general solution of the following: () () () 9π Solve the following roblems: () ; () ; () () 5 ; () ; () Solve the following boundar value roblems: π () + ; () ; () π + + ; () ; () 8 ; ( ) ; () e Find a general solution: () + () + π () + + Solve the following differential equations: () () + () ( D D D + ) () (D D + 9) Solve the following initial value roblem: () ; () ; () ; () () + ; () () () () ( D + D + 9) ; () () () () Advanced Engineering Mathematics b Erwin Kreszig..

8 Year - Tutorial: 8 Toic: Modeling Euler Cauch Equations k Show that the harmonic oscillation ( t) Acosω t + B sinωt; ω Starting m from initial dislacement with initial velocit v is v ( t) t cosω + ( sinωt ω ) B and reresents this in the form ( t) C cos( ω t δ ); C A + B tanδ. A If a sring is such that a weight of nt (about.5lb) would stretch it cm what would the frequenc of the corresonding harmonic oscillation be? The eriod? How does the frequenc of the harmonic oscillation change if we (i) double the mass (ii) take a stiffer sring? Before ou look at formulas first tr to find qualitative answers b hsical arguments. Could ou make a harmonic oscillation move faster b giving the bod a greater initial ush? 5 ( α β ) t ( α + β ) t Show that for ( t) ce + ce to satisf initial conditions ( ) and α + v ( ) v we must have β β c and + v c α β v Show that in the over damed case the bod can ass through at most case (see fig. age no. 88) 9 7 αt Find the critical motion ( t) ( c + ct) e that starts from with initial velocit v 9 8 αt Under what conditions does ( t) ( c + ct) e have a maimum or minimum at some instant t >? 9 Find a real general Solution in E (i) + +. (ii) β Advanced Engineering Mathematics b Erwin Kreszig..

9 Year - Tutorial: 9 Toic: Euler Cauch Equations Eistence and Uniqueness Theor. Wronskian Find a general solution : ( D + D + ). 9 Find a general solution : ( D +.5) 9 Solve the following initial value roblems in E () () 9 ( D D + ) () () 9 Find the wronskian of the given bases in E. 5 9 and verif Theorem (age no. 98) 5 e e a / a / e cos e sin 7 ln 8 µ µ cos( ln ) sin( ln ) Find a second order homogenous linear differential equation for which given functions are solutions. Find the wronskian and use it to verif linear indeendence. 9 e e ln. Advanced Engineering Mathematics b Erwin Kreszig..

10 Year - Tutorial: Toic: Non Homogeneous equation Verif that is a solution equation of the given differential equation and find a general solution 8e e ( D + D ) 8cos + sin cos ( D + ) + lnπ lnπ Verif that is a solution of the given equation. Solve the initial value roblem. 5 + () () 8; e () () 7 ( D + ) sin ().8 () 5.; cos 8 ( D D + ) ln () () ; ln e 9 ( D D + ) ( + ) e () + e () e; e D D e ( + + ) () () e ln e e. Advanced Engineering Mathematics b Erwin Kreszig..

11 Year - Tutorial: Toic: Solutions b Undetermined coefficient Find a (real) general solution. Which rules are ou using? 7 + sin + 8 cosh e + 7sin e cosh Solve the given initial value roblem. Indicate the rules ou are using () () 8 9 e () () e () () Tutorial: Toic: Non Homogeneous Equation Find a general solution of following + / / + / / cos 9 + / Advanced Engineering Mathematics b Erwin Kreszig..

12 Year - Tutorial: Toic: Non Homogeneous Equation Find the stead-state oscillation of the mass- sring sstem governed b the given equation Find the transient motion of the mass-sring sstem governed b the given equation ; 7 Find the motion of the mass-sring sstem corresonding to the given equation and initial conditions. State the time when the solution racticall reaches the stead state ; ; 7 Prove that the given functions form a basis of the corresonding given equation. Then solve the initial value roblem. 7 ; 8 ; ; + ; + ; 8. Advanced Engineering Mathematics b Erwin Kreszig..

13 Year - Tutorial: Toic: Higher order Homogeneous equation with constant coefficient Solve the following differential equation 7 iv iv + + (D D + 9) Solve the following initial value roblems and lot the solution 7 5 iv () () () () + () () () 7 ( D + 5D D 5) () 5 () () 5 8 ( D + D + 8D + 8D + ) () () () () Find a General solution e e Solve the following initial value roblems " e sin () () () 5 + () () (). Advanced Engineering Mathematics b Erwin Kreszig..

14 Year - Tutorial: 5 Toic: Series Solution Al the ower series method to the following differential equations. Show the details of our work Find a ower series solution in owers of of the following differential equations. 5 k( const.) + Series solution in owers of can be obtained if ou introduce t as a new indeendent variable and solve the resulting equation for as a ower series in t. Do this for the following equations with. Show details of our work. 7 k 8 + Determine the radius of convergence of the following series. m 9 ( ) m m m m! m. Advanced Engineering Mathematics b Erwin Kreszig.. Tutorial: Toic: Series Solution Find a basis of solutions of the following differential equations. Show the details of our work. Tr to identif the series as eansions of known functions. ( ) + ( ) ( ) + ( ) ( + ) + ( + ) 5 ( ) ( + ) Find a general solution in terms of her geometric functions ( ) + ( ) 9 ( ) ( ) + ( + )

15 Year - Tutorial: 7 Toic: Lalace Transforms Find the Lalace transform of the following functions 57 t () f ( t) e cosh 5t () f ( t) sin t cos t () f ( t) 5sin t Find the Lalace transform of the following functions 57 t 5t t () f ( t) t e () f ( t) ( t + ) e () f ( t) e cos t Find the inverse Lalace transform of the following functions 5.s +.9 ak S + S 8 ( ) F ( S) () F ( S) s () F( S) 5 +. S K S S 57 k + Find the inverse Lalace transform of the following functions () F ( S) 5 () S F ( S) ( S ) ( S + ) π 5 Show that L t e d S > It is ossible to show that e d 5 Give simle eamle of functions have no Lalace Transform. Indicate the reason. 7 Solve the following initial value roblem b the Lalace transform () + 5sint () () + a a () () t () + e () () 8 Let f ( t) t sin wt. Find L{f(t)} etend the method Of differentiation in above eamle to S w obtain L{ t cos wt} From this derive: ( S + w ) ( ) L ( S w ) + w ) (sin wt wt cos wt) S ( L t sin wt ( S + w ) w 9 From above eamles obtain similar formula for Herbolic functions S + a as () L{ t cosh at} () L{ t sinh at} ( S a ) ( S a ) Find f (t). [Theorem: New inverse transforms b Integration] ( ) F( S) () 5 π F( S) ) F( S) ( S + S) S ( S + π ) S( S + w 9 S + F ( S) () S ( S + 9) ( ) 5. Advanced Engineering Mathematics b Erwin Kreszig..

16 No. Year - Tutorial: 8 Toic: Lalace Transforms. Advanced Engineering Mathematics b Erwin Kreszig.. 7 Alication of the second shifting theorem find Lalace Transforms of the following 7 () f ( t) tu( t ) π sin wt; < t < t () f ( t) e u( t ) () f ( t) w π π ; < t < w w Find and sketch the inverse Lalace transform s 5s πs () ( e e ) e e s ( ) ( ) s s + s + ( s + π ) Using Lalace transform solve the following roblems 7 t; < t < : () () () + 7 ; () ().5 () : t. > () u( t ) + δ ( t ) : () ; () Find Lalace transforms of the following eamles b Convolution. 8 at ( ) e bt * e t () u( t )* e ( )cos wt * coswt 5 Find f(t) b the convolution theorem 8 as () F( s) ) ( ) () ( ) s( s + ) s e F s F s ( s + w ) s( s ) Aling convolution Find and sketch the solution 8 () + cost : () ; () < t < Where r ( t) () + + r( t) t > 7 Using Lalace transform solve the integral equations 8 8 t + ( ) ( t) ( u) du ( ) t ( u)( t u) du ( ) ( t) sin t + ( u)sin ( t u) du Show that forced vibro eress t f ( t) t t + : < t < t e : t < 5 t sin t : t 5 As a linear combination of Heaviside s ste functions 9 Solve the given initial value roblem b Lalace transforms ) + : + : () () ( ( ) + : : () () Sketch the grah of the given function and eress in terms Of the unit ste function u(t) t < ( ) f ( t) ( t ) e : t : () f ( t ) 7 t t : t < : 5 t < 5 t < 7 : t 7 t

17 Year -. Advanced Engineering Mathematics b Erwin Kreszig..