For any differentiable function y of x , d^ 2 x d y^ 2 ((d y)/(d x))^ 3 + d^ 2 y d x^ 2 =

Correct answer: A. 0 (d y)/(d x)=((d x)/(d y))^ -1 d^ 2 y d x^ 2 =-1((d x)/(d y))^ -2 \ (d)/(d x)((d x)/(d y))\ d^ 2 y d x^ 2 =(-1)((d x)/(d y))^ -2 \ (d)/(d y)((d x)/(d y)) (d y)/(d x)\ =(-1)((d y)/(d x))^ 2 \ d^ 2 x d y^ 2 · (d y)/(d x)\ =-((d y)/(d x))^ 3 \ d^ 2 x d y^ 2 \ d^ 2 x d y^ 2 ((d y)/(d x))^ 3 + d^ 2 y d x^ 2 =0

BITSAT 2016

Exam: BITSAT
Year: 2016
Subject: Mathematics
Chapter: Differential Equations

BITSAT 2016 · Mathematics · Differential Equations

For any differentiable function y of x , d^ 2 x d y^ 2 ((d y)/(d...

For any differentiable function

y

x

\frac{d^{2} x}{d y^{2}}\left(\frac{d y}{d x}\right)^{3}+\frac{d^{2} y}{d x^{2}}=

Options

A0
B $y$
C $-y$
D $x$

Correct answer

A. 0

Step-by-step solution

\frac{d y}{d x}=\left(\frac{d x}{d y}\right)^{-1}\Rightarrow \frac{d^{2} y}{d x^{2}}=-1\left(\frac{d x}{d y}\right)^{-2}\left\{\frac{d}{d x}\left(\frac{d x}{d y}\right)\right\}\Rightarrow \frac{d^{2} y}{d x^{2}}=(-1)\left(\frac{d x}{d y}\right)^{-2}\left\{\frac{d}{d y}\left(\frac{d x}{d y}\right) \frac{d y}{d x}\right\}=(-1)\left(\frac{d y}{d x}\right)^{2}\left\{\frac{d^{2} x}{d y^{2}} \cdot \frac{d y}{d x}\right\}=-\left(\frac{d y}{d x}\right)^{3}\left\{\frac{d^{2} x}{d y^{2}}\right\}\Rightarrow \frac{d^{2} x}{d y^{2}}\left(\frac{d y}{d x}\right)^{3}+\frac{d^{2} y}{d x^{2}}=0

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