Q1/ What is the relationship between Laplace transform 00 L[u(t)] = F(s) = e-stu(t)dt 0. and the following transform -t E[u(t)] = T(w) = w ew u(t)dt Some Properties:- 1. E[1] = w² 2. E[t] = w³ 3. E[eax] = w2 1-aw Q2/ Use RDTM to solve the following nonlinear system: = u2v 2u+ +1 (4+1 Ju(x,y,t) at av(x,y,t) at with initial conditions 1/0²u =u-u²v+1 (+ 4x2 u(x, y, 0) = e-x-y v(x, y, 0) = ex+y J² u ay² a²u) ay2 Q3/ Find Adomian polynomial A6 for the nonlinear term F(u). Q4/Use homotopy perturbation method to solve the following Helmholtz equation: a²u(x, y) a²u(x, y) ax² + with initial conditions Jy2 - u(x, y) = 0, u(o,y) =y, ux(0,y) = y + coshy.

Q: When calculing the Laplace transform of the function f(t)=e^2t.sin(4t) is obtained as a result:

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Q: Question 5 The following function is homogenous of degree what? Vx?y²+x²+y² f(x,y) = х, у ху 1 3 4…

A: We have to find the degree of function.

Q: 6. Find the inverse Laplace transform of the function (3²-²)(as + b) I(s) = (s² + a²) (s² + 3²)

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Q: What is the inverse Laplace Transform of this function if one term is 1/s and the other is 1/s+a?

A: The inverse Laplace transform of F(s) is f(t) = 3 + 4e^-2t - 7e^-5t

Q: . Find the Laplace transform of the function f(t) = 2t + 3 cos(3t) + 4 sin(3t)

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Q: The Laplace transform of x(t) is X(s). Given x(0)=2 and x'(0)=-1, write an expression for the…

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Q: Find the inverse Laplace transform L-{F(s)} of the function e-2s F(s) : %D s2 + 5s – 50 NOTE:…

A: Given that Fs=e-2ss2+5s-50 We have to find the inverse Laplace transform of Fs that is L-1Fs For…

Q: = Find the inverse Laplace transform f(t) = £¹{F(s)} of the function F(s) = f(t) = = L 6s+2 s² + 100…

A: The function: Fs=6s+2s2+100. We have to find the inverse Laplace transform of the given function.

Q: 2. Let 2s6 s³3s2 + 4s - 12 Find the inverse Laplace transform, y(s). Y(s):

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Q: Find the inverse Laplace transform f(t) = L¹{F(s)} of the function F(s) = = f(t) = L -- {7+5)-1…

A: Solution:-

Q: 5. Find the inverse Laplace transform of the function H(s) = as + b (s-a)² +32° a = 9,b= 10, a = 4,8…

A: We have to find the inverse Laplace transform of the function H(t)=as+b(s-α)2+β2 when a=9, b=10,…

Q: y -7y +10y= 9cost+7sint, 5, y =-4 2

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