Question

2 (a) Use the method of separation of variables to derive the solution of the following intial-boundary-value problem: $u_{tt}(x, t) - c^2u_{xx}(x,t) = 0$, $0 < x < L$, $0 < t$, $u(0,t) = u(L,t) = 0$, $0 le t$, $u(x, 0) = phi(x)$, $u_t(x, 0) = psi(x)$, $0 le x le L$, where $c$ is a given positive constant and $phi$, $psi$ are given functions. (b) What assumptions would you make about the functions $phi$, $psi$ in part (a) in order for your solution to part (a) to be valid? (c) Write down the solution $u(x, t)$ to part (a) when the functions $phi$, $psi$ are given by the following formulas: $phi(x) = 0$, $psi(x) = 4sinleft(frac{2pi}{L}x ight)$. (d) Use standard trigonometric identities to write the solution $u(x, t)$ that you found in part (c) in the form $u(x,t) = f(x-ct) + g(x+ct)$ where $f$ and $g$ are a suitable pair of functions.

          2 (a) Use the method of separation of variables to derive the solution of the following
intial-boundary-value problem:
$u_{tt}(x, t) - c^2u_{xx}(x,t) = 0$, $0 < x < L$, $0 < t$,
$u(0,t) = u(L,t) = 0$, $0 le t$,
$u(x, 0) = phi(x)$, $u_t(x, 0) = psi(x)$, $0 le x le L$,
where $c$ is a given positive constant and $phi$, $psi$ are given functions.
(b) What assumptions would you make about the functions $phi$, $psi$ in part (a) in order
for your solution to part (a) to be valid?
(c) Write down the solution $u(x, t)$ to part (a) when the functions $phi$, $psi$ are given
by the following formulas:
$phi(x) = 0$, $psi(x) = 4sinleft(frac{2pi}{L}x
ight)$.
(d) Use standard trigonometric identities to write the solution $u(x, t)$ that you found
in part (c) in the form
$u(x,t) = f(x-ct) + g(x+ct)$
where $f$ and $g$ are a suitable pair of functions.

$2 (a) Use the method of separation of variables to derive the solution of the following intial-boundary-value problem: utt(x, t) - c^2uxx(x,t) = 0, 0 < x < L, 0 < t, u(0,t) = u(L,t) = 0, 0 le t, u(x, 0) = phi(x), ut(x, 0) = psi(x), 0 le x le L, where c is a given positive constant and phi, psi are given functions. (b) What assumptions would you make about the functions phi, psi in part (a) in order for your solution to part (a) to be valid? (c) Write down the solution u(x, t) to part (a) when the functions phi, psi are given by the following formulas: phi(x) = 0, psi(x) = 4sinleft(frac2piLx ight). (d) Use standard trigonometric identities to write the solution u(x, t) that you found in part (c) in the form u(x,t) = f(x-ct) + g(x+ct) where f and g are a suitable pair of functions.$

Added by Tanya M.

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