Scalar integrating factor for a reaction-diffusion equation ($v_t=v_{xx}+a(t)v,v=e^{\int a(t)dt}u$)

A spatially uniform time-dependent reaction coefficient can be removed from a linear diffusion equation by a scalar integrating factor. If $A^{\prime }(t)=a(t)$ 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.