If the general solutions of a differential equation is $(y + c)^{2} = c x$ , where $c$ is an arbitrary constant, then the order and degree of differential equation are:

1, 2

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2, 1

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1, 1

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None of these

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Solution

The correct option is D $1, 2$
Given general solution is ${(y + c)}^{2} = c x$
Now differentiate it with respect to $x$ on both sides
we get $2 (y + c) \frac{d y}{d x} = c$ , Let $\frac{d y}{d x}$ be $Y$
$\Rightarrow c = \frac{2 y Y}{1 - 2 Y}$
By substituting $c$ in general solution , we get ${(y + \frac{2 y Y}{1 - 2 Y})}^{2} = \frac{2 y Y}{1 - 2 Y} x$
$\Rightarrow y^{2} = 2 y x Y (1 - 2 Y) = 2 y x Y - 4 y x Y^{2}$
Therefore the order of equation is $1$ and the degree of equation is $2$
Therefore option $A$ is correct

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