Q9 a) Find the Fourier Transformation of f(x) = b) Find the Fourier series expansion of f(x) = 0, x < 0 -x² x>0 1+2: if -/ < x < 0 if 0/
Q9 a) Find the Fourier Transformation of f(x) = b) Find the Fourier series expansion of f(x) = 0, x < 0 -x² x>0 1+2: if -/ < x < 0 if 0/
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve Q9
![Q3 a) Solve the following initial value problem using Laplace transformation
d²y dy
+4 +8y= cos 2t with y(0) = 2,y (0) = 1
dt²
dt
Show that L(5²)=√(-1/4
b)
Q4 a) Verify Green's Theorem in the plane for f (xy+y²)dx + x²dy, where 'C' is
the closed curve of the region bounded by y= x² and y = x
b)
Find the area bounded by one arch of the cycloid x= a(0-sin0), y =
a(1- cos 0); 0≤ 0 ≤ 2π
Q5 a) Prove that the integral
b) Prove that r(-) = 2√
105
sin a cos ax
-dw =
2* 2 * 2 ** 2 (upto 'K' times) =
Q6 a) Solve the following integral equation using Laplace transformation
y(t) = 1 + cos(t-u)y(u)du
b) Using convolution prove that
2K₂K-1
(K-1)!
;0<x< 1
x = 1
x > 1
4
0;
;
Q7 a)
Verify Stokes Theorem, when F= yî + (x-2xz)j-xyk and surface 'S' Is the
part of the sphere x² + y² + z² = a² above the x-y plane.
b)
Find the total Mass of a mass distribution of density f(x, y, z)= e-x-y-z in a
region T: 0≤ x ≤ 1-y,0 ≤ y ≤ 1,0 ≤z≤2
Q9 a) Find the Fourier Transformation of f(x) =
Q8 a) Verify Divergence Theorem for F= zî + xĵ-yzk taken over the surface of the
cylinder x² + y² = 9 included in the first octant between
z = 0 and z = 4
b) Find the coordinates of the center of gravity of a mass of density
f(x, y) = 1 in the region R: the triangle with vertices (0,0), (b, 0) and (h)
b) Find the Fourier series expansion of f(x) =
So, x < 0
le-x², x > 0
1+2x
2
1-2x
if -/< x < 0
if 0<x</](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff297114c-ed17-40e3-bb40-78b98ac3f9da%2F23545c64-c97d-4524-bb49-f7fbc8d55d79%2Fj2pwa4h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q3 a) Solve the following initial value problem using Laplace transformation
d²y dy
+4 +8y= cos 2t with y(0) = 2,y (0) = 1
dt²
dt
Show that L(5²)=√(-1/4
b)
Q4 a) Verify Green's Theorem in the plane for f (xy+y²)dx + x²dy, where 'C' is
the closed curve of the region bounded by y= x² and y = x
b)
Find the area bounded by one arch of the cycloid x= a(0-sin0), y =
a(1- cos 0); 0≤ 0 ≤ 2π
Q5 a) Prove that the integral
b) Prove that r(-) = 2√
105
sin a cos ax
-dw =
2* 2 * 2 ** 2 (upto 'K' times) =
Q6 a) Solve the following integral equation using Laplace transformation
y(t) = 1 + cos(t-u)y(u)du
b) Using convolution prove that
2K₂K-1
(K-1)!
;0<x< 1
x = 1
x > 1
4
0;
;
Q7 a)
Verify Stokes Theorem, when F= yî + (x-2xz)j-xyk and surface 'S' Is the
part of the sphere x² + y² + z² = a² above the x-y plane.
b)
Find the total Mass of a mass distribution of density f(x, y, z)= e-x-y-z in a
region T: 0≤ x ≤ 1-y,0 ≤ y ≤ 1,0 ≤z≤2
Q9 a) Find the Fourier Transformation of f(x) =
Q8 a) Verify Divergence Theorem for F= zî + xĵ-yzk taken over the surface of the
cylinder x² + y² = 9 included in the first octant between
z = 0 and z = 4
b) Find the coordinates of the center of gravity of a mass of density
f(x, y) = 1 in the region R: the triangle with vertices (0,0), (b, 0) and (h)
b) Find the Fourier series expansion of f(x) =
So, x < 0
le-x², x > 0
1+2x
2
1-2x
if -/< x < 0
if 0<x</
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