Solve the following equation for y(t) using Fourier Transforms. dy(t) dt - +2y(t) = e¯¹h(t) where h(t) is the Heaviside function: [0,t < 0 1,t ≥ 0 Note: the solution satisfies ly(t) →0 as t→±∞0. h(t) =

Q: Use the chain rule to find dw dt dw dt W = In (√x² + y² + z²), x = 9 sin(t), y = 8 cos(t), z = 5…

A: W=lnx2+y2+z2 x=9 sin(t) y=8 cos(t) z=5 tan(t)

Q: Determine the Laplace transform of the solution to Jy" - 4y + 2y = g(t) [y(0) = 8, y'(0) = 0 where…

A: The given differential equation is:y''-4y'+2y=g(t)y(0)=8,  y'(0)=0where:g(t)=0…

Q: Use the Laplace transform to solve the initial value problem: 0St<3 (-5, y(0) = 0, y'(0) =-1 y +9y=…

A:

Q: . Let r(t) = (t², 1 t, 4t). Calculate the derivative of r(t) · a(t) at t = 6, assuming that a(6) =…

A:

Q: Find the formula for the Laplace transform of the derivative of a function y'. Use the initial…

A: The solution is given as follows

Q: Find the extremal for the following functional with x(1) = 1 and x(2) = 10. ²x² (t) J -1² dt 2t3

A:

Q: A2 Use the Fourier Transform Method to find the solution u(x,t) of the initial value problem: ut +…

A: Solve for u(x,t): We now have:U(ω, t) = F(ω) * e^(-iωt^2/2)Invert the Fourier Transform to obtain…

Q: Find  d/dt [ w(t) ]^2 and show that it equals 2w(t)w'(t)

A: Answer

Q: Suppose that the function y(t) satisfies the DE y" + y = sin 3t, with initial values, y(0) = 2,…

A:

Q: Let h(t) = (t + √ t − 1 )t . Find  ∫h(t) dt.

A: Given : ht = t  + t -1t

Q: C. Time Differentiation. Determine the Laplace Transform of y(t) given all the initial conditions…

A: We have to solve the given problem by Laplace transform technique.

Q: d² dy +5 dy + 5y = e-t sint, where y(0) = 0 and y' (0) = 1.

A: Given: The equation is d2ydt2+5dydt+5y=e-tsint. The given initial condition is y0=0 and y'0=1. To…

Q: (b) Find x(1) by using Laplace transform and inverse Laplace method of following differential…

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Q: Use the Laplace transform to solve the given initial-value problem. y"+y= √2 sin(√2t), y(0) 13,…

A: Solution:-

Q: Determine the Laplace transform of the differential equation. y² + 4y y (0) = 0 = Cost;

A: Sol:- y' +4y=costTake Laplace transform of both sides of the equationLy'…

Q: Use an appropriate Fourier Transform Theorem to solve the following Integral Equation ƒ(t) +6 [**°…

A:

Q: Use the Fourier Transform Method to find the solution u(x,t) of the initial value problem: ut + tux…

A: Step 1: Apply Fourier Transform to the PDE The Fourier transform u_hat(k,t) of u(x,t) is defined by…

Q: Use Laplace transforms to solve the initial value problem where primes indicate derivatives with…

A: I am going to solve the problem by using some simple algebra to get the required result of the given…