Question

Dy/DX= ay- x^2/y^2- ac, find d^2y/dx^2

Answer 1

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$\implies\displaystyle\bf{\dfrac{Dy}{Dx}=\dfrac{ay-x^2}{y^2-ac}}$

$\red\bigstar\underline{\bf{ Double \: Differentiating \: the \: following \: Equation}}$

✪By using Quotient Rule :

$\: \: \\ \large\implies \displaystyle \bf{ \frac{ {d}^{2} y}{ {dx}^{2} } = \frac{ {y}^{2} - ac \frac{dy}{dx}(ay - {x}^{2} ) - \frac{dy}{dx} ( {y}^{2} - ac) \times ay - {x}^{2} }{ {( {y}^{2} - ac }^{2}) } }$

$\: \\ \implies \bf{ \frac{ {d}^{2}y }{ {dx}^{2} } = \frac{ {y}^{2} - ac(a \frac{dy}{dx} - 2x - (2y - 0) \times ay - {x}^{2} }{ {( {y}^{2} - ac })^{2} } }$

$\: \\ \implies \displaystyle \bf{ \frac{ {d}^{2} y}{ {dx}^{2} } = \frac{ {y}^{2}(a - 2x) - ac(a - 2x) - 2y(ay - {x}^{2} )}{ {( {y}^{2} - ac})^{2} } }$

$\: \\ \implies \displaystyle \bf{ \frac{ {d}^{2}y }{ {dx}^{2} } = \frac{ {y}^{2}a - 2x {y}^{2} - {a}^{2}c + 2xac - 2a {y}^{2} + 2y {x}^{2} }{ ({ {y}^{2} - ac})^{2} } }$

$\: \\ \implies \bf{ \frac{ {d}^{2} y}{ {dx}^{2} } = \frac{ - {y}^{2}a - 2x {y}^{2} - {a}^{2}c + 2xac + 2yx }{ {( {y}^{2} - ac })^{2} } }$