Question

If y^2= ax^2 + b, then d^2y/dx^2​

Answer 1

Answer:

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Answer 2

Answer:

The required second order differential is

$\frac{d {}^{2}y }{dx {}^{2} } = \frac{ab}{y {}^{3} }$

Step-by-step explanation:

The given expression is

$y {}^{2} = ax {}^{2} + b$

Differentiating with respect to x of both sides,we get

$2y. \frac{dy}{dx} = 2ax \\ = > y. \frac{dy}{dx} = ax$

Redifferentiating with respect to x on both sides,we get

$y. (\frac{d {}^{2}y }{dx {}^{2} } )+ ( \frac{dy}{dx} )( \frac{dy}{dx} ) = a \\ = > y. (\frac{d {}^{2}y }{dx {}^{2} } )+ ( \frac{dy}{dx} ) {}^{2} = a$

We know that,

$\frac{dy}{dx} = \frac{ax}{y}$

Therefore,the relation becomes

$y. (\frac{d {}^{2}y }{dx {}^{2} } )+ (\frac{ax}{y}) {}^{2} = a \\ = > \frac{d {}^{2} y}{dx {}^{2} } = \frac{a - (\frac{ax}{y} ) {}^{2} }{y} = \frac{ay {}^{2} - a {}^{2} x {}^{2} }{y {}^{3} } \\ = \frac{a(ax {}^{2} + b) - a {}^{2}x {}^{2} }{y {}^{3} } = \frac{ab}{y {}^{3} }$

Therefore the required second order differential is

$\frac{d {}^{2}y }{dx {}^{2} } = \frac{ab}{y {}^{3} }$

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