It is given that $\mathrm{e}^y = 2 + \mathrm{e}^x$. Show that \(\frac{\mathrm{d}^2y}{\ma

Question

It is given that $\mathrm{e}^y = 2 + \mathrm{e}^x$. Show that $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 = \frac{\mathrm{d}y}{\mathrm{d}x}$. Hence find the Maclaurin series for $y$ up to and including the term in $x^2$, giving the coefficients in exact form.[4] By using appropriate standard series expansions from the List of Formulae (MF26), verify the correctness of the series expansion found in part (a).[3]

Accepted Answer

(a) $\ln 3+\frac{x}{3}+\frac{{{x}^{2}}}{9}+\ldots $(b) $\ln 3+\frac{x}{3}+\frac{{{x}^{2}}}{9}+\ldots $ (verified)

2023 DHS Promo Q4

2023 DHS Promo Q4