1. The problem statement, all variables and given/known data If there is heat radiation in a rod of length L, then the 1-D heat equation might take the form: u_t = ku_xx + F(x,t) exercise deals with the steady state condition => temperature u and F are independent of time t and that u_t = 0. u_t = partial derivative with respect to t u_xx = 2nd partial derivative with respect to x Problem: find u(x) if F(x) = sinx, k = 2, u(0) = u'(0), u(L) = 1 2. Relevant equations e^(itheta) = cos(theta) + isin(theta) e^(-itheta) = cos(theta) - isin(theta) e^(itheta) + e^(-itheta) = 2cos(theta) e^(itheta) - e^(-itheta) = 2isin(theta) 3. The attempt at a solution u_t = ku_xx + F(x,t) u_t = ku_xx + F(x)G(t) 2u_xx + sin(x)G(t) = 0 characteristic equation: m^2+1=0, m^2 = -1, m = +-i general soltn: u(x) = c1cos(x) + c2sin(x) u(0) = c1cos(0) + c2sin(0) u'(x) = -c1sin(x) + c2cos(x) u'(0) = -c1sin(0) + c2cos(0) => c2 = c1 u(x) = c1cos(x) + c1sin(x) u(L) = c1cos(L) + c1sin(L) = 1 from here I'm actually a little lost, I'm not sure what to do with u(L) = 1 portion; do I perhaps need to find a particular solution? any help would be appreciated.