I have been trying to find out how to solve the following problem, but to no avail. Can anybody tell me what to do, hint where to start, or anything that might help?
Find the real eigensolutions to the damped heat equation $u_t=u_{xx}-u$.
2026-03-31 18:45:01.1774982701
PDE - Real Eigensolutions to the Damped Heat Equation
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Perform separation of variables: $$ u_t +u = u_{xx} \\ u(t,x)=T(t)X(x) \\ \frac{T'}{T}+1 = \lambda,\;\; \lambda=\frac{X''}{X} \\ T(t) = Ce^{(\lambda-1)t},\;\;\; X(x)=A\sin(\sqrt{\lambda}x)+B\cos(\sqrt{\lambda}x) $$ You'll need conditions in $x$ to go any further.