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Differential Calculus> find d 2 y/dx 2 (independent of t) of the...

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find d²y/dx² (independent of t) of the function defined paramaterically as x = sin (ln t) and y = cos (ln t)

Satyadip Mahapatra , 9 Years ago

Grade 12

1 Answers

BALAJI ANDALAMALA

Last Activity: 9 Years ago

x = sin(lnt) and y=cos(lnt)

$x^2+y^2 = sin^2(lnt)+cos^2(lnt) = 1$
The equation satisfies the above two parametric equations is x^2+y^2=1.

Differentiating with respect to x , we get
$2x+2y\frac{dy}{dx}=0.$
$\Rightarrow \frac{dy}{dx}=\frac{-x}{y}............(1)$
Again differentiating equation (1) with respect to x, we get

$\frac{d^2y}{dx^2}=\frac{-1-(\frac{dy}{dx})^2}{y} = \frac{-1-(\frac{-x}{y})^2}{y}$
$\therefore \frac{d^2y}{dx^2}= = \frac{-(y^2+x^2)}{y^3} = \frac{-1}{y^3}$ $(\because \,\,x^2+y^2=1)$

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