The differential equation obtained by elimina | vedclass

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Question

(C) Given equation: $y^2 = (x + c)^3$ Differentiating both sides with respect to $x$: $2y \frac{dy}{dx} = 3(x + c)^2$ From this,we have $(x + c)^2 = \

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$8\left(\frac{dy}{dx}\right)^3 = 27y$

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(C) Given equation: $y^2 = (x + c)^3$ Differentiating both sides with respect to $x$: $2y \frac{dy}{dx} = 3(x + c)^2$ From this,we have $(x + c)^2 = \frac{2y}{3} \frac{dy}{dx}$. Cubing both sides: $(x + c)^6 = \left(\frac{2y}{3} \frac{dy}{dx}ight)^3$ Since $(x + c)^3 = y^2$,we have $(x + c)^6 = (y^2)^2 = y^4$. Substituting this into the equation: $y^4 = \frac{8y^3}{27} \left(\frac{dy}{dx}ight)^3$ Dividing both sides by $y^3$ (assuming $y 
eq 0$): $y = \frac{8}{27} \left(\frac{dy}{dx}ight)^3$ $27y = 8 \left(\frac{dy}{dx}ight)^3$

The differential equation obtained by eliminating the arbitrary constant from the equation $y^2 = (x + c)^3$ is

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