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A system is described by the inhomogeneous partial differential equation Sxx + 27,+/- = 41e-x/2sin( 3x) with boundary conditi

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Answer #1

so here we have given that an inhomogeneous equation and we have to find the solution f(t,x) with the help of

DUHAMEL PRINCIPLE WHICH relates the solution of the given equation with auxiliary solution

and we have given already the Auxiliary equation .so here we dont need to find it

we just find the solution f(t,x) with the help of this formula given below

so f(t,x)=\int_{0}^{t} w(t,x,s) ds

=2e37t/8-x/2 sin3x  \int_{0}^{t}(s.e-37s/8)ds

now we solve with help of integration by parts which is stated below

{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)dx\\[6pt]&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)dx.\end{aligned}}}

=2 e37t/8-x/2 sin3x [s{e-37s/8 (-8/37)}+ 8/37\int_{0}^{t} e-37s/8 ds]  

=2 e37t/8-x/2 sin3x [s{e-37s/8(-8/37)} + 8/37 {(-8/37)e-37s/8}]

taking limit 0 to t in complete integral where you can see the closed bracket [ ]

=2.e37t/8-x/2 sin3x [(-8/37t.e-37t/8)-(64/1369.e-37t/8+ 64/1369)]

= e-x/2 sin3x [ -16/37 t - 128/1369 + 128/1369 e37t/8]

= 16/1369 .e-x/2 sin3x [-37t-8+8.e37t/8]

so option (C) is the correct answer

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