Consider the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x} + y \tan x = 2 \cos^4 x$ given

Question

Consider the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x} + y \tan x = 2 \cos^4 x$ given that $y = 1$ when $x = 0$. Solve the differential equation, giving your answer in the form $y = \mathrm{f}(x)$. Using differentiation, show that $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + y = -10 \sin x \cos^3 x$. Hence find the first $4$ terms of the Maclaurin series for $y$.

Accepted Answer

(a) $y=\cos x\left(1+2\sin x-\dfrac23\sin^3x\right)$; (b) $y''+y=-10\sin x\cos^3x$; (c) $y=1+2x-\dfrac12x^2-2x^3+\cdots$

Solution of Differential Equations using Maclaurin's series

Solution of Differential Equations using Maclaurin's series

Solution:

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