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# Differential equation of all circles radius 5, with their centres on the y-axis.

$(a)\;x^2 (\frac{dx}{dy})^2+x^2+25=0 \\ (b)\;x^2 (\frac{dy}{dx})^2+x^2+25=0 \\ (c)\;x^2 (\frac{dx}{dy})^2-x^2+25=0 \\ (d)\;y^2(\frac{dx}{dy})^2-x^2+25$

Let centre (0,h)
then $(x-0)^2+(y-h)^2=(5)^2$
$x^2-25=-(y-h)^2$
$25-x^2=(y-h)^2$
$-2x =2 (y-h)\large\frac{dy}{dx}$
$25-x^2=-(-x \large\frac{dx}{dy})^2$
$25-x^2 = - x^2 (\large\frac{dx}{dy})^2$
$+ x^2 (\frac{dx}{dy} )^2 -x^2 +25=0$
Hence c is the correct answer.