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Problem 4. A string is stretched between the points x = 0 and x = 1 and is initially at rest along the x-axis. Its motion due to gravity is described by the solution u(x, t) of the following problem: ? utt = c^2 uxx - g, 0 < x < 1, t > 0 { u(0, t) = u(1, t) = 0, t > 0 ? u(x, 0) = ut(x, 0) = 0, 0 < x < 1 Here the constant g is the gravitational acceleration. (i) Find the time-independent solution v(x) of the above equation which satisfies the homogeneous boundary conditions as well. (ii) Show that w(x, t) = u(x, t) - v(x) satisfies the wave equation wtt = c^2 wxx. Find the corresponding boundary and initial conditions for w and use the solution available to you to express w(x, t) as a Fourier series. (iii) Use (i) and (ii) to find a formula for u(x, t).

          Problem 4. A string is stretched between the points x = 0 and x = 1 and is initially at rest along the x-axis. Its motion due to gravity is described by the solution u(x, t) of the following problem:

? utt = c^2 uxx - g, 0 < x < 1, t > 0
{ u(0, t) = u(1, t) = 0, t > 0
? u(x, 0) = ut(x, 0) = 0, 0 < x < 1

Here the constant g is the gravitational acceleration.
(i) Find the time-independent solution v(x) of the above equation which satisfies the homogeneous boundary conditions as well.
(ii) Show that w(x, t) = u(x, t) - v(x) satisfies the wave equation wtt = c^2 wxx. Find the corresponding boundary and initial conditions for w and use the solution available to you to express w(x, t) as a Fourier series.
(iii) Use (i) and (ii) to find a formula for u(x, t).

Problem 4. A string is stretched between the points x = 0 and x = 1 and is initially at rest along the x-axis. Its motion due to gravity is described by the solution u(x, t) of the following problem:

? utt = c^2 uxx - g, 0 < x < 1, t > 0
u(0, t) = u(1, t) = 0, t > 0
? u(x, 0) = ut(x, 0) = 0, 0 < x < 1

Here the constant g is the gravitational acceleration.
(i) Find the time-independent solution v(x) of the above equation which satisfies the homogeneous boundary conditions as well.
(ii) Show that w(x, t) = u(x, t) - v(x) satisfies the wave equation wtt = c^2 wxx. Find the corresponding boundary and initial conditions for w and use the solution available to you to express w(x, t) as a Fourier series.
(iii) Use (i) and (ii) to find a formula for u(x, t).

Added by Alicia A.

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