Shri G. S.
Institute of Institute of Technology and Science
Department of AppliedMathematics and Computational Science
B.Tech. II Year: ELEX & Telecommunication Engineering (July-Dec 2024)
MA-25014:MATHEMATICS-III, ASSIGNMENT-II
Last Date of Submission: 14-10-2024
1. 4 x , 3 x 4 CO2
(i) Find the Fourier series of f ( x) .
x 4 , 4 x 5
x
(ii) Find Fourier series of f ( x ) in ( 0, 2 ) and hence deduce that
2
1 1 1
1 ... .
4 3 5 7
2. Analyze harmonically the data given below and express y = f(x) in Fourier series up to the CO2
third harmonic
x 0 π/3 2 π/3 π 4 π/3 5 π/3 2π
y 1.0 1.4 1.9 1.7 1.5 1.2 1.0
3. (i) Using Fourier Integral representation, show that CO2
cos (𝜆𝑥) 𝜋
𝑑𝜆 = 𝑒 , (𝑥 > 0)
(1 + 𝜆 ) 2
0 , x 0
1
(ii)Find Fourier Integral representation of f ( x ) , x 0 .
2
e x , x 0
4. (i) Write Fourier Sine & Cosine Transform. CO2
ax
(ii)Find Fourier sine transform and Fourier cosine transform of f ( x ) e ,a 0 .
5. CO2
Solve u 2 u2 , if u (0, t ) 0 , u ( x ,0) e x , x 0 , u ( x , t )
2
is bounded
t x
where x 0 , t 0 .
6. (i) Find Laplace transform of the following functions: CO3
2 2
t
sin 2t
(a) (1 te t )3 (b) cos(t ) if t (c) t 2 sin t (d) (e) e t t cos t dt (f)
3 3 t 0
t ,0 t b
f (t ) with period 2b (g) t 2 u (t 1) (t 1)
2b t , b t 2b
(ii) State and prove convolution theorem.
7. Find Inverse Laplace transform of the following functions: CO3
5s 3 s2 s
(a) (b) 2 (c) cot 1 .
( s 1)( s 2 s 5)
2
s ( s 1)( s 2) 2
8. Using Laplace transformation method to solve the following equations: CO3
d 2x dx dx
(a) 2 2 x et given that x 2, 1 at t 0 .
dt dt dt
dx dy
(b) y et , x sin t given that x (0) 1, y (0) 0 when t 0 .
dt dt
9. 0 ,t 0 CO3
Define unit step function and unit impulse function .Express function f ( t ) sin t , 0 t
0 ,t
in unit step function and find its Laplace transform.
10. s CO3
Using convolution theorem to evaluate L1 2 .
( s 1)( s 4)
2
11. (i)Find a real root of the equation using Newton Raphson method x 2 x cos x 0 . CO5
(ii) Use Picard’s method to approximate the value of y when x =0.1 given that y=1 when x=0 ,
dy
3x y 2
dx
12. (i)Solve the system of equations 3 x 4 y 5 z 18 , 2 x y 8 z 13 , 5 x 2 y 7 z 20 using CO5
Gauss elimination method .
(ii) Find a root of the equation x3 – 4x – 9 = 0, using the bisection method correct to three
decimal places.
13. (i)Solve the equations 10 x 2 y z 9, x 10 y z 22 , 2 x 3 y 10 z 22 by Gauss CO5
Seidal method.
x
(ii)Using Regulafalsi method, compute the real root of the equation xe = 2, correct to four
decimal places.
14. dy 1 CO5
Find the value of y (0.5) using RungeKutta method, given that , y (0) 1
dx x y
15. dy CO5
Apply Taylor’s series method to find the value of y (0.2) , given that 2 y 3e x , y (0) 0.
dx
Compare the numerical solution with the exact solution.
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