The variables $x$ and $y$ are related by the differential equation $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = 2\mathrm{e}^{-2x} .$ Solve the differential equation. Hence find a possible solution curve that has an oblique asymptote, and state the equation of the oblique asymptote.

1. Find the first derivative by integrating the given second derivative with respect to $x$.

$\frac{\mathrm{d}y}{\mathrm{d}x} = \int 2\mathrm{e}^{-2x} \, \mathrm{d}x = -\mathrm{e}^{-2x} + C$

2. Find the general solution by integrating the first derivative.

$y = \int \left( -\mathrm{e}^{-2x} + C \right) \, \mathrm{d}x = \frac{1}{2}\mathrm{e}^{-2x} + Cx + D$
where $C$ and $D$ are arbitrary constants.

3. Deduce a specific possible solution curve that possesses an oblique asymptote.

As $x \to \infty$, $\mathrm{e}^{-2x} \to 0$, so $y \to Cx + D$.
For the curve to have an oblique asymptote, it must approach a straight line with a non-zero gradient, meaning we must choose $C \neq 0$.
Letting $C = 1$ and $D = 0$ gives a valid possible solution curve:
$y = \frac{1}{2}\mathrm{e}^{-2x} + x$

4. State the equation of the corresponding oblique asymptote.

For the chosen solution curve $y = \frac{1}{2}\mathrm{e}^{-2x} + x$, the curve approaches the line $y = x$ as $x \to \infty$.
Thus, the equation of the oblique asymptote is:
$y = x$