Help with this PDE problem

Question

Given the equation $au_{xx}(x,t)-bu_{t}(x,t)+cu(x,t)=0$

define $w(x,t)=e^{\delta t}u(x,t)$ and find the equation satisfied by $w$.

Could you guys give me a hint on how to proceed, or a reference that can help me with this?

Thanks in advance

Answer 1

I'm sorta curious to see how this works out, so:

With

$au_{xx}(x, t) - bu_t(x, t) + cu(x, t) = 0, \tag 1$

and

$w(x, t) = e^{\delta t}u(x, t), \tag 2$

we have, per the suggestion in the comment of Mattos,

$u(x, t) = e^{-\delta t} w(x, t), \tag 3$

from which

$u_{xx} = e^{-\delta t} w_{xx}(x, t), \tag 4$

$u_t(x, t) = -\delta e^{-\delta t} w(x, t) + e^{-\delta t} w_t(x, t); \tag 5$

we assemble (3)-(5) into (1):

$a e^{-\delta t} w_{xx}(x, t) - b(-\delta e^{-\delta t} w(x, t) + e^{-\delta t} w_t(x, t)) + c e^{-\delta t} w(x, t)$ $= e^{-\delta t}(a w_{xx}(x, t) + b\delta w(x, t) - bw_t(x, t) + c w(x, t))$ $= e^{-\delta t}(aw_{xx}(x, t) - bw_t(x, t) + (c + \delta b)w(x, t)); \tag 6$

it now follows by virtue of (1) that

$e^{-\delta t}(aw_{xx}(x, t) - bw_t(x, t) + (c + \delta b)w(x, t)) = 0, \tag 7$

whence, upon multiplication by $e^{\delta t}$,

$aw_{xx}(x, t) - bw_t(x, t) + (c + \delta b)w(x, t) = 0. \tag 8$