Q.(vi) The differential equation representing the family of circles $x^2+(y-a)^2=a^2$ will be of order two. (State True or False.)

The family $x^2+(y-a)^2=a^2$ has just one arbitrary constant $a$. The order of the differential equation representing a family equals the number of independent arbitrary constants, so eliminating $a$ needs only one differentiation and gives a first-order equation. …

The key principle

The order of the differential equation that represents a family of curves equals the number of independent arbitrary constants in the family, because each constant needs one differentiation to eliminate it.

The family

contains a single arbitrary constant $a$ (it fixes both the centre $(0,a)$ and the radius $a$ at once). So we expect order one. Let us confirm it.

Eliminate the constant

1. Simplify. Expand:

2. Differentiate once with respect to $x$:

3. Remove $a$. From step 1, $a=\dfrac{x^2+y^2}{2y}$. Substituting,

Multiply by $2y$ and collect the derivative terms: …

Method: Order of the DE Representing a Family of Curves

The order of the differential equation that represents a family equals the number of independent arbitrary constants in the family — count them first, before differentiating.

Steps

Step 1: Count the essential arbitrary constants.

Look at the family and identify how many independent parameters it truly contains (not how many times a letter appears).

Step 2: Differentiate just enough times.

To eliminate $n$ constants you differentiate $n$ times; the resulting equation has order $n$.

Step 3: Eliminate the constant(s). …

Common Mistakes

Mistake 1: Counting $a$ and $a^2$ as two different constants.

Why it's wrong: $x^2+(y-a)^2=a^2$ contains a single parameter $a$; $a^2$ is not independent of $a$. Correct approach: count independent arbitrary constants — here just one — so the order is one.

Mistake 2: Assuming a "circle" family must give a second-order equation. …