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Question

Form the differential equation corresponding to y2 − 2ay + x2 = a2 by eliminating a.

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Solution

The equation of the family of curves is
y2-2ay+x2=a2 ...(1)
where a is a parameter.
This equation contains only one arbitrary constant, so we shall get a differential equation of first order.
Differentiating equation (1) with respect to x, we get
2ydydx-2adydx+2x=02ydydx+2x=2adydxy+xdydx=a
Substituting the value of a in equation (2), we get
y2-2y+xdydxy+x2=y+xdydx2y2dydx-2ydydx+xy+x2dydxdydx=ydydx+x2dydx2y2dydx2-2y2dydx2-2xydydx+x2dydx2=y2dydx2+2xydydx+x2x2-2y2dydx2-4xydydx-x2=0 It is the required differential equation.

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