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Question Number 8488 by Basant007 last updated on 12/Oct/16
find y_n  if y=cos 2x
$${find}\:{y}_{{n}} \:{if}\:{y}=\mathrm{cos}\:\mathrm{2}{x} \\ $$
Commented by FilupSmith last updated on 13/Oct/16
y_n =(d^n y/dx^n )  y_1 =(1/2)sin(2x)  y_2 =−(1/4)cos(2x)  y_3 =(1/8)sin(2x)  y_4 =−(1/(16))cos(2x)     ∴y_n =(−1)^(n+1) (1/2^n )×???  working /// thinking
$${y}_{{n}} =\frac{{d}^{{n}} {y}}{{dx}^{{n}} } \\ $$$${y}_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\left(\mathrm{2}{x}\right) \\ $$$${y}_{\mathrm{2}} =−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{cos}\left(\mathrm{2}{x}\right) \\ $$$${y}_{\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{8}}\mathrm{sin}\left(\mathrm{2}{x}\right) \\ $$$${y}_{\mathrm{4}} =−\frac{\mathrm{1}}{\mathrm{16}}\mathrm{cos}\left(\mathrm{2}{x}\right) \\ $$$$\: \\ $$$$\therefore{y}_{{n}} =\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}^{{n}} }×??? \\ $$$$\mathrm{working}\:///\:\mathrm{thinking} \\ $$
Commented by Rasheed Soomro last updated on 13/Oct/16
What relation is there between  y_(n  ) and  y=cos 2x ?
$$\mathrm{What}\:\mathrm{relation}\:\mathrm{is}\:\mathrm{there}\:\mathrm{between} \\ $$$${y}_{{n}\:\:} \mathrm{and}\:\:{y}=\mathrm{cos}\:\mathrm{2}{x}\:? \\ $$
Commented by Basant007 last updated on 13/Oct/16
find using succesive differentiation
$${find}\:{using}\:{succesive}\:{differentiation} \\ $$
Commented by Basant007 last updated on 13/Oct/16
i.e y_n
$${i}.{e}\:{y}_{{n}} \\ $$
Commented by FilupSmith last updated on 13/Oct/16
y_n = { (((1/2^n )sin(2x)          if 2∣n)),((−(1/2^n )cos(2x)     if 2∤n)) :}
$${y}_{{n}} =\begin{cases}{\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\mathrm{sin}\left(\mathrm{2}{x}\right)\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:\mathrm{2}\mid{n}}\\{−\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\mathrm{cos}\left(\mathrm{2}{x}\right)\:\:\:\:\:\mathrm{if}\:\mathrm{2}\nmid{n}}\end{cases} \\ $$
Commented by Basant007 last updated on 13/Oct/16
i tried and got this
$$\mathrm{i}\:\mathrm{tried}\:\mathrm{and}\:\mathrm{got}\:\mathrm{this} \\ $$
Commented by Basant007 last updated on 13/Oct/16
y_n =a^n cos {n(π/2)+2x}
$$\mathrm{y}_{\mathrm{n}} =\mathrm{a}^{\mathrm{n}} \mathrm{cos}\:\left\{\mathrm{n}\frac{\pi}{\mathrm{2}}+\mathrm{2x}\right\} \\ $$

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