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) $y^{\prime\prime}+(y^\prime)^2=y^\prime$ and $y=\ln3+\frac x3+\frac{x^2}{9}+O(x^3)$. (b) The standard-series calculation gives the same expansion.

2023 DHS Promo Q4

2023 DHS Promo Q4

Solution:

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