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(1 point) Solve the nonhomogeneous heat problem Ut = uzz + sin(4x), 0 < x < , u(0,t) = 0, u(1,t) = 0 u(x,0) = 5 sin(3x) u(x,

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If the non homogeneous heat problem is of the form, Uz- & Uxx = 0(2,7) ocacl 0 ) ozco) with conditions U(0, 2) = UCI, z)=0 u( Comparing the coefficients in equation 2 and O 2=1, IET, $(x,x) = sin(4x) and F(X) = 5 Singse The solution of ② is given by uUsing the result, Ŝs Sin conx) sin (ox) dx = f O, if mfn Tl, if nan Thus, Sin (3x) Sin (2x) dx = O, if on 73 if n-3 Hence, En=) Fn (s) = Lo if n74 l, if n=4 from ③ we have + Ucxx) = Ñ En Sincome est Sin conx) S いい -n²z toeg . F. (3) ds nel z -97 5) Ulin t-Go U(0,7) = 0 + sin(4x) (1-0) 16 .. lim re(a,t) = 1 Sin(4x) 16

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