## Teaching Scheme (in Hours)

Theory Tutorial Practical Total
3 2 0 5

Subject Credit :  5

Theory
ESE (E)
Theory
PA (M)
Practical
ESE Viva (V)
Practical
PA (I)
Total
0

## Syllabus Content

###### Unit-1:  Introduction to Some Special Functions

Gamma function, Beta function, Bessel function, Error function and complementary Error function, Heaviside’s function, pulse unit height and duration function, Sinusoidal Pulse function, Rectangle function, Gate function, Dirac’s Delta function, Signum function, Saw tooth wave function, Triangular wave function, Halfwave rectified sinusoidal function, Full rectified sine wave, Square wave function.

###### Unit-2:  Fourier Series and Fourier integral

Periodic function, Trigonometric series, Fourier series, Functions of any period, Even and odd functions, Half-range Expansion, Forced oscillations, Fourier integral

###### Unit-3:  Ordinary Differential Equations and Applications

First order differential equations: basic concepts, Geometric meaning of y’ = f(x,y) Direction fields, Exact differential equations, Integrating factor, Linear differential equations, Bernoulli equations, Modeling , Orthogonal trajectories of curves.

Linear differential equations of second and higher order: Homogeneous linear differential equations of second order, Modeling: Free Oscillations, Euler- Cauchy Equations, Wronskian, Non homogeneous equations, Solution by undetermined coefficients, Solution by variation of parameters.

Modeling: free Oscillations resonance and Electric circuits, Higher order linear differential equations, Higher order homogeneous with constant coefficient, Higher order non homogeneous equations. Solution by [1/f(D)] r(x) method for finding particular integral.

###### Unit-4:  Series Solution of Differential Equations

Power series method, Theory of power series methods, Frobenius method.

###### Unit-5:  Laplace Transforms and Applications

Definition of the Laplace transform, Inverse Laplace transform, Linearity, Shifting theorem, Transforms of derivatives and integrals Differential equations, Unit step function Second shifting theorem,Dirac’s delta function, Differentiation and integration of transforms,Convolution and integral equations, Partial fraction differential equations, Systems of differential equations

###### Unit-6:  Partial Differential Equations and Applications

Formation PDEs, Solution of Partial Differential equations f(x,y,z,p,q) = 0, Nonlinear PDEs first order, Some standard forms of nonlinear PDE, Linear PDEs with constant coefficients,Equations reducible to Homogeneous linear form, Classification of second order linear PDEs.Separation of variables use of Fourier series, D’Alembert’s solution of the wave equation,Heat equation: Solution by Fourier series and Fourier integral

## Reference Books

Sr. Title Author Publication Amazon Link
1 Advanced Engineering Mathematics (8th Edition) E. Kreyszig Wiley-India (2007)
2 Engineering Mathematics Vol 2 Baburam Pearson
3 Elementary Differential Equations (8th Edition) W. E. Boyce and R. DiPrima John Wiley (2005)
4 Fourier series and boundary value problems (7th Edition) R. V. Churchill and J. W. Brown McGraw-Hill (2006)
5 Calculus , Volume-2 ( 2nd Edition ) T.M.Apostol Wiley Eastern , 1980

## Course Outcome

After learning the course the students should be able to:

1. Fourier Series and Fourier Integral
• Identify functions that are periodic. Determine their periods.
• Find the Fourier series for a function defined on a closed interval.
• Find the Fourier series for a periodic function.
• Recall and apply the convergence theorem for Fourier series.
• Determine whether a given function is even, odd or neither.
• Sketch the even and odd extensions of a function defined on the interval [0,L].
• Find the Fourier sine and cosine series for the function defined on [0,L]
2. Ordinary Differential Equations and Their Applications
• Model physical processes using differential equations.
• Solve basic initial value problems, obtain explicit solutions if possible.
• Characterize the solutions of a differential equation with respect to initial values.

### BE (CE) ⇒ Semester: 3

Darshan Institute of Engineering & Technology is a leading institute offering undergraduate (B.E.), postgraduate (M.E.) and Diploma programs in engineering.

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