$\begin{array}{1 1}(A)xyy'' + x(y')^2 - y' = 0 \\(B)xyy'' - x(y')^2 - yy' = 0 \\(C)xyy'' + x(y')^2 + yy' = 0 \\ (D)xyy'' + x(y')^2 - yy' = 0 \end{array} $

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- The equation of a hyperbola having foci on $x$-axis and centre at origin is $\large\frac{x^2}{a^2} -\frac{ y^2}{b^2}$$ = 1$

Step 1:

From the information in the tool box we take the equation of the hyperbola as $\large\frac{x^2}{a^2} -\frac{ y^2}{ b^2 }$$= 1$-----------(1)

Differentiating this on both sides we get,

$\large\frac{2x}{a^2} - [\frac{1}{b^2}]$$[(2y'y'] = 0$

on simplifying we get $\large\frac{x}{a^2} -(\frac{yy'}{b^2})$ = 0------(2)

Step 2:

Again differentiating this we get

$\large\frac{1}{a^2} - (\frac{1}{b^2})$$ [ (y')^2 + yy''] = 0$

$\large\frac{1}{a^2} = [\frac{1}{b^2}]$$[(y')^2 + yy'']$

Substituting this in equ (2) we get

$\large\frac{x}{b^2}$$[(y')^2 + yy''] - \large\frac{yy'}{b^2}$$ = 0$

Step 3:

Multiplying by $b^2$ throughout and simplifying we get

$xyy'' + x(y')^2 - yy' = 0$

This is the required differential equation.

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