# Engineering Mathematics 3 Solved Problems (1) Which of the following sets are linearly independent and which are linearly dependent?(a) (b) (c) (d) (e) (f) (2) For each of the following differential equations, write down the differential operator, L, that will allow the equation to be expressed in the form L[x(t)] = 0.Hence write down the general solution of the differential equation (a) d 2 x dt 2 −p 2 x= 0 dx dt d 2 x dt 2 dx dt x= 0 dx dt (1) The temperature, T, (in Kelvin) of a reaction vessel during a particularly unstable chemical reaction is modelled by the linear differential equation, d 4 T dt 4 d 3 T dt 3 d 2 T dt 2 d T dt At the start of the experiment the reactor vessel is at 293K.

(1) Which of the following sets are linearly independent and which are linearly dependent?(a) (b) (c) (d) (e) (f) (2) For each of the following differential equations, write down the differential operator, L, that will allow the equation to be expressed in the form L[x(t)] = 0.Hence write down the general solution of the differential equation (a) d 2 x dt 2 −p 2 x= 0 dx dt d 2 x dt 2 dx dt x= 0 dx dt (1) The temperature, T, (in Kelvin) of a reaction vessel during a particularly unstable chemical reaction is modelled by the linear differential equation, d 4 T dt 4 d 3 T dt 3 d 2 T dt 2 d T dt At the start of the experiment the reactor vessel is at 293K.

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However, 2 is the least value and therefore is the period of f(x). The Fourier series of an even function in (-l , l) contains only cosine terms (constant term included), i.e. Com Transforms and Partial differential equations I year / I sem = where 2. Find the Fourier series of period 2 for the function = x cos x in 0 50Prepared by : P.

Com Transforms and Partial differential equations I year / I sem CHAPTER 1 FOURIER SERIES A function is said to have a period T if for all x, , where T is apositive constant. EXAMPLES 1.1 We know that = sin x = sin (x 4 ) = Therefore the function has period 2 ,4 , 6 , etc. Com Transforms and Partial differential equations I year / I sem where n is an integer where n is an integer2.6.1 PROBLEMS1.

 SCEE08009 Engineering Mathematics 2A Tutorial 2 (d) ( d 3 x dt 3 t d 2 x dt 2 x 2 t= sint, x(1) = 1,x ̇(1) = 0,x ̈(1) =− 2. 2 ,x ̇(1) = 1,x ̈(1) = 0, using forward Euler with a step size of ∆t= 0.025s.

Repeat the calculation with ∆t= 0.0125 and hence estimate the accuracy of your solution att= 2.

For tutoring resources, visit missed midterm or final exam may be made up; however, it is the student’s responsibility to establish with documentation that the exam was missed for a solid reason.

The student cannot make up a missed midterm or final exam without such documentation.Regular attendance at the lectures and the labs is expected.It is the student’s responsibility to know what is going on in class.(a) dx dt −kx= 0 (b) d 3 x dt 3 t 2 dx dt =t d dt (xt) (d) d dt t d dt (t 2 x) =xt.(3) Determine which members of the given sets are solutions of the differential equation.UNIT II FOURIER TRANSFORM 9Cosine transforms Properties (without Proof) Transforms of simple functions Convolutiontheorem Parsevals identity Finite Fourier transform Sine and Cosine transform. S., Higher Engineering Mathematics, Thirty Sixth Edition, Khanna Publishers, Delhi, 2001. Kandasamy, P., Thilagavathy, K., and Gunavathy, K., Engineering Mathematics Volume III, S. Z-transform - Elementary properties (without proof) Inverse Z transform Convolution theorem -Formation of difference equations Solution of difference equations using Z - transform. K., Integral Transforms for Engineers and Applied Mathematicians, Macmillen , New York ,1988. For more information, see University Regulation 4.001 at compliance with the Americans with Disabilities Act (ADA), students who require special accommodation due to a disability to properly execute coursework must register with the Office for Students with Disabilities (OSD) -- in Boca Raton, SU 133 (561-297-3880) and follow all OSD procedures.UNIT I FOURIER SERIES 9Fourier series Odd and even functions Half range sine series Half range cosine series Complexform of Fourier Series Parsevals identify Harmonic Analysis. A function defined in c x c 2l can be expanded as an infinite trigonometric 1. is continuous or piecewise continuous with finite number of finite discontinuities in (c , c 2l). has no or finite number of maxima or minima in (c , c 2l). Com Transforms and Partial differential equations I year / I sem1.4 DEFINITION OF FOURIER SERIESFourier series of in the interval c x c 2l, provided the coefficients are given by the Eulers formulas. ODD FUNCTION If = in (-l , l) such that = - , then is said to be an oddfunction of x in (-l , l). the Fourier series of an odd function in (-l , l) is given by = , where1.6 PROBLEMS1. x = l lies in (0 , 2l) and is a point of continuity of the function = x(2l x). W., Fourier Series and Boundary Value Problems, Fourth Edition, Mc Graw-Hill Book Co., Singapore, 1987. Similarly cos x is a periodic function with the period 2 and tan x has period . the Fourier series of an even function in (-l , l) is given by Prepared by : P. The Fourier series of an odd function in (-l , l) contains only sine terms, i.e. Com Transforms and Partial differential equations I year / I sem Using these values in (1), we haveseries in (2).

## Comments Engineering Mathematics 3 Solved Problems

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