The differential equation for the family

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Target Exam

CUET

Subject

-- Mathematics - Section A

Chapter

Differential Equations

Question:

The differential equation for the family of curves $x^2+y^2-2ay=0$ where a is an arbitrary constant is:

Options:

$(x^2-y^2)y'=2xy$

$2(x^2+y^2)y'=xy$

$2(x^2-y^2)y'=2xy$

$(x^2+y^2)y'=2xy$

Correct Answer:

$(x^2-y^2)y'=2xy$

Explanation:

$x^2+y^2-2ay=0$ … (i)

Differentiate w. r. t. x ⇒ 2x + 2py - 2ap = 0 $⇒a=\frac{2x+2py}{2p}$ (where p = dy/dx)

Substitute the value of a in (i)

$x^2+y^2-y[\frac{2x}{p}+2y]=0$

$⇒(x^2-y^2)p=2xy$