= Scalar integrating factor for a reaction-diffusion equation
{title2=$v_t=v_{xx}+a(t)v,\quad v=e^{\int a(t)dt}u$}
A spatially uniform time-dependent reaction coefficient can be removed from a linear <diffusion equation> by a scalar <integrating factor>. If $A'(t)=a(t)$ and $v=e^{A(t)}u$, substitution gives $u_t=u_{xx}$. Initial data and boundary values must also be divided by $e^{A(t)}$. A nonzero boundary profile can then be removed by <boundary lifting> before applying <separation of variables>. This reduction distinguishes growth of the whole amplitude from decay of the spatial diffusion modes.
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