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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 2* 2 * 2 ** 2 (upto 'K' times) = (K-1)!

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 2* 2 * 2 ** 2 (upto 'K' times) = (K-1)!

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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</
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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