Techniques of Differentiation - Second and Higher Derivatives

Using the techniques of differentiation, or otherwise, answer the following questions:

1. Given that \(y = \frac{x^2}{\mathrm{e}^x}\) and \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + 2\frac{\mathrm{d}y}{\mathrm{d}x} = m \mathrm{e}^{-x} - y\), find the value of \(m\).
2. Given that \(y = a \mathrm{e}^{2x} + b \mathrm{e}^{-x}\) and \(\frac{\mathrm{d}y}{\mathrm{d}x} + 4y = \mathrm{e}^{2x} - \mathrm{e}^{-x}\), find the values of \(a\) and \(b\).