1. Suppose $u(t, x)$ solves the equation
$\frac{\partial}{\partial t} u = \frac{\partial^2}{\partial x^2} u + 2u$.
Let $v(t, x) = e^{-2t}u(t, x)$. Compute $\frac{\partial}{\partial t} v$, $\frac{\partial^2}{\partial x^2} v$ in terms of $\frac{\partial}{\partial t} u$, $\frac{\partial^2}{\partial x^2} u$ and then show that $v(t, x)$ solves the heat equation $\frac{\partial}{\partial t} v = \frac{\partial^2}{\partial x^2} v$.
