2026 ASRJC P1 Q7

Given that \(y=e^{\sqrt{1-x}}\), show that \[2\ln y\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+\frac2y\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2=-\frac{\mathrm{d}y}{\mathrm{d}x}.\][2]

1. By further differentiation of this result, or otherwise, find the Maclaurin's series of \(y\) up to and including the term in \(x^3\).[4]
2. Deduce the series expansion of \(\displaystyle y=\frac{e^{\sqrt{1-x}}}{\sqrt{1-x}}\) up to and including the term in \(x^2\).[2]