Given that $y = \tan^{-1} \frac{1}{5}x$, show that \((25 + x^2) \frac{\mathrm{d}^2y}{\mathrm{d}x^2

Question

Given that $y = \tan^{-1} \frac{1}{5}x$, show that $(25 + x^2) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 2x \frac{\mathrm{d}y}{\mathrm{d}x} = 0$.[2] By further differentiation of this result, find the Maclaurin's series of $y$ up to and including the term in $x^3$.[3]

Accepted Answer

(a) Identity shown. (b) $y=\frac{x}{5}-\frac{x^3}{375}+\cdots$.

2025 ASRJC P2 Q3

2025 ASRJC P2 Q3

Solution:

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