Math  /  Calculus

Question3 It is given that, at any point on the graph of y=f(x),dy dx=exsinxyy=\mathrm{f}(x), \frac{\mathrm{d} y}{\mathrm{~d} x}=\mathrm{e}^{x} \sin x-y. (i) Show that d3y dx3=d2y dx22y\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}=\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 y.
The graph of y=f(x)y=\mathrm{f}(x) passes through the origin OO. (ii) Find the Maclaurin series for yy, up to and including the term in x4x^{4}. [4] (iii) Hence, find the Maclaurin series for e2xsin2x\mathrm{e}^{2 x} \sin 2 x, up to and including the term in x2x^{2}. [2]

Studdy Solution
The Maclaurin series for yy is x22+x36+x424+\frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \dots.
The Maclaurin series for e2xsin2x\mathrm{e}^{2x} \sin 2x is 2x+10x23+2x + \frac{10x^2}{3} + \dots

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