# The differential equation whose solution is $\dpi{100} Ax^{2}+By^{2}=1,\; where\; A\; and\; B$ are arbitrary constants is of Option 1) second order and second degree Option 2) first order and second degree Option 3) first order and first degree Option 4) second order and first degree

As we learnt in

Formation of Differential Equations -

A differential equation can be derived from its equation by the process of differentiation and other algebraical process of elimination

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$Ax^{2}+ By^{2}= 1$

$\Rightarrow 2Ax+2By\frac{dy}{dx}= 0$

$\Rightarrow Ax+By\frac{dy}{dx}= 0$

$\Rightarrow A.1+B\left[\frac{dy}{dx}.\frac{dy}{dx}+y.\frac{d^2 y}{dx^2} \right ] = 0$

$\Rightarrow A+B\left[ \left(\frac{dy}{dx} \right )^{2}+y\frac{d^{2}y}{dx^{2}}\right ]=0$

By eliminating A and B,

$xyy''+x(y')^{2}- yy'=0$

Option 1)

second order and second degree

Incorrect option

Option 2)

first order and second degree

Incorrect option

Option 3)

first order and first degree

Incorrect option

Option 4)

second order and first degree

Correct option

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