Question

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b:y2= a(b2-x2)

Answer 1

y2= a(b2-x2)
Differentiating both sides with respect to x, we get:

$2y\frac{dy}{dx}=a(-2x)$

$\Rightarrow 2yy'=-2ax$

$\Rightarrow yy'=-ax$...(1)
Again, differentiating both sides with respect to x, we get:
y'.y'+yy''=-a
$\Rightarrow (y')^2+yy''=-a$...(2)
Dividing equation(2)by equation(1),we get:
$(y')^2+\frac{yy''}{y'}=-\frac{a}{-ax}$
$\Rightarrow xyy''+x(y')^2-yy''=0$

This is the required differential equation of the given curve.