Change of variables for heat equation

Question

How to make a change of variables to turn the equation $$\frac{\partial{u}}{\partial{t}}=D\frac{\partial^2{u}}{\partial{x}^2}+cu$$ back to the heat equation? Where can I read about change of variables? Thank you.

Answer 1

Hint: use the change of variables $u = \exp{(\alpha t + \beta x)} \cdot v (x, t)$. Your task is to plug this expression and guess $\alpha$ and $\beta$ such that they remove all terms except $v_t$ and $v_{xx}$.

Answer 2

$\dfrac{\partial u}{\partial t}=D\dfrac{\partial^2u}{\partial x^2}+cu$

$\dfrac{\partial u}{\partial t}-cu=D\dfrac{\partial^2u}{\partial x^2}$

$e^{-ct}\dfrac{\partial u}{\partial t}-ce^{-ct}u=De^{-ct}\dfrac{\partial^2u}{\partial x^2}$

$\dfrac{\partial(e^{-ct}u)}{\partial t}=D\dfrac{\partial^2(e^{-ct}u)}{\partial x^2}$

$\therefore$ Let $v=e^{-ct}u$ , the PDE becomes $\dfrac{\partial v}{\partial t}=D\dfrac{\partial^2v}{\partial x^2}$ .