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Question:

If the differential equation representing the family of all circles touching x-axis at the origin is (x²−y²)dy/dx=g(x)y, then g(x) equals:

12x

2x

12x²

2x²

Solution:

(x²−y²)y′=g(x)y
x²+(y−a)²=a²
Differentiating the equation with respect to x
2x+2(y−a)y′=0
a=x+yy′/y′ (1)
Put 'a' in original equation, we get
x²+(y−(x+yy′/y′))²=(x+yy′/y′)²
y′²x²+(yy′−(x+yy′²))²=(x+yy′)²
By solving this equation
(x²−y²)y′=2xy
g(x)=2x