(a) Let $q(x, t)$ satisfy the heat equation

$$\frac{\partial q}{\partial t}=\frac{\partial^{2} q}{\partial x^{2}}$$

Find the function $X$, which depends linearly on $\partial q / \partial x, q, k$, such that the heat equation can be written in the form

$$\frac{\partial}{\partial t}\left(e^{-i k x+k^{2} t} q\right)+\frac{\partial}{\partial x}\left(e^{-i k x+k^{2} t} X\right)=0, \quad k \in \mathbb{C} .$$

Use this equation to construct a Lax pair for the heat equation.

(b) Use the above result, as well as the Cole-Hopf transformation, to construct a Lax pair for the Burgers equation

$$\frac{\partial Q}{\partial t}-2 Q \frac{\partial Q}{\partial x}=\frac{\partial^{2} Q}{\partial x^{2}}$$

(c) Find the second-order ordinary differential equation satisfied by the similarity solution of the so-called cylindrical $\mathrm{KdV}$ equation:

$$\frac{\partial q}{\partial t}+\frac{\partial^{3} q}{\partial x^{3}}+q \frac{\partial q}{\partial x}+\frac{q}{3 t}=0, \quad t \neq 0$$