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find d 2 y/dx 2 (independent of t) of the function defined paramaterically as x = sin (ln t) and y = cos (ln t)

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

Grade:12

1 Answers

BALAJI ANDALAMALA
askIITians Faculty 78 Points
6 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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