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Question: The differential equation representing the family of hyperbola a<sup>2</sup>x<sup>2</sup> –b<sup>2</...

The differential equation representing the family of hyperbola a2x2 –b2y2 = c2 is –

A

y′′y′\frac{y^{''}}{y^{'}} + y′y\frac{y^{'}}{y} = 1x\frac{1}{x}

B

y′′y′\frac{y^{''}}{y^{'}} + y′y\frac{y^{'}}{y} = 1x2\frac{1}{x^{2}}

C

y′′y′\frac{y^{''}}{y^{'}} – y′y\frac{y^{'}}{y} = 1x\frac{1}{x}

D

y′′y′\frac{y^{''}}{y^{'}} = yy′\frac{y}{y^{'}} – 1x\frac{1}{x}

Answer

y′′y′\frac{y^{''}}{y^{'}} + y′y\frac{y^{'}}{y} = 1x\frac{1}{x}

Explanation

Solution

Differentiating the equation twice w.r.t. x, we have

2a2 x – 2b2yy' = 0, a2 – b2(y'2 + yy'') = 0

Eliminating a2 and b2 we have the differential equation

y′′y′+y′y=1x\frac{y^{''}}{y^{'}} + \frac{y^{'}}{y} = \frac{1}{x}