First slide

Formation of differential equations

Question

The equation of the family of curves which intersect the hyperbola xy=2 orthogonally is

Easy

A

$y= \frac{x^{3}}{6} +c$
B

$y= \frac{x^{2}}{4} +c$
C

$y= \frac{{-x}^{2}}{4} +c$
D

$y= \frac{{-x}^{3}}{6} +c$

Solution

$y=2/x$
$\begin{array}{l} \frac{dy}{dx} = \frac{- 2}{x^{2}} \\ Let y = d (x) be the required family of curves then {(\frac{dy}{dx})}_{c_{1}} {(\frac{dy}{dx})}_{c_{2}} = - 1 \\ \to \frac{d y}{d x} x - \frac{2}{x^{2}} = - 1 \to \frac{d y}{d x} = \frac{x^{2}}{2} \\ \Rightarrow y = \frac{x^{3}}{6} + c \end{array}$

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