Form the differential equation by eliminating A and B in Ax2 + By… - Vidyadip

Question

Form the differential equation by eliminating A and B in Ax² + By² = 1.

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Answer

Given equation is Ax² + By² = 1
On differentiating both sides w.r.t.x, we get
$2\text{Ax}+2\text{By}\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow2\text{By}\frac{\text{dy}}{\text{dx}}=-2\text{Ax}$
$\Rightarrow\text{By}\frac{\text{dy}}{\text{dx}}=-\text{Ax}$
$\Rightarrow\frac{\text{y}}{\text{x}}.\frac{\text{dy}}{\text{dx}}=-\frac{\text{A}}{\text{B}}$
Again, differentiating w.r.t.x, we get
$\frac{\text{y}}{\text{x}}.\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+\frac{\text{dy}}{\text{dx}}.\bigg(\frac{\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}}{\text{x}^2}\bigg)=0$
$\Rightarrow\frac{\text{y}}{\text{x}}.\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+\frac{\text{x}\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-\text{y}\Big(\frac{\text{dy}}{\text{dx}}\Big)}{\text{x}^2}=0$
$\Rightarrow\text{xy}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)^2-\text{y}\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)=0$
$\Rightarrow\text{xy}\text{y}''+\text{x}(\text{y}')^2-\text{y}\text{y}'=0$

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