Question

Consider the below wave equation with the given conditions.\ $36 frac{partial^2 u}{partial x^2} = frac{partial^2 u}{partial t^2}$, $0 < x < 7$, $t > 0$,\ $u(0, t) = u(7, t) = 0$, $t > 0$\ $u(x, 0) = 0$, $left. frac{partial u}{partial t} ight|_{t = 0} = 4x(7 - x) = sum_{n=1}^{infty} frac{784}{pi^3 n^3} {1 - (-1)^n} sin(frac{npi x}{7})$, $0 < x < 7$.\ The solution to the above boundary-value problem is of the form\ $u(x, t) = sum_{n=1}^{infty} g(n, t) sin(frac{npi x}{7})$\ Find the function $g(n, t)$.

          Consider the below wave equation with the given conditions.\
$36 frac{partial^2 u}{partial x^2} = frac{partial^2 u}{partial t^2}$, $0 < x < 7$, $t > 0$,\
$u(0, t) = u(7, t) = 0$, $t > 0$\
$u(x, 0) = 0$, $left. frac{partial u}{partial t} 
ight|_{t = 0} = 4x(7 - x) = sum_{n=1}^{infty} frac{784}{pi^3 n^3} {1 - (-1)^n} sin(frac{npi x}{7})$, $0 < x < 7$.\
The solution to the above boundary-value problem is of the form\
$u(x, t) = sum_{n=1}^{infty} g(n, t) sin(frac{npi x}{7})$\
Find the function $g(n, t)$.

$Consider the below wave equation with the given conditions.36 fracpartial^2 upartial x^2 = fracpartial^2 upartial t^2, 0 < x < 7, t > 0,u(0, t) = u(7, t) = 0, t > 0u(x, 0) = 0, left. fracpartial upartial t ight|t = 0 = 4x(7 - x) = sumn=1^infty frac784pi^3 n^31 - (-1)^n sin(fracnpi x7), 0 < x < 7.The solution to the above boundary-value problem is of the formu(x, t) = sumn=1^infty g(n, t) sin(fracnpi x7)Find the function g(n, t).$

Added by Nathan S.

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