Q.Form the differential equation representing the family of curves $y^2 = m(a^2 - x^2)$ by eliminating the arbitrary constants $m$ and $a$.

The key idea is Eliminating Arbitrary Constants. To form a differential equation from a family of curves, we differentiate the given equation repeatedly until all arbitrary constants are eliminated. Since there are two arbitrary constants ($m$ and $a$), we will differentiate twice.

1. Differentiate the given equation once:

   The given family of curves is $y^2 = m(a^2 - x^2)$ (1).

   Differentiating (1) with respect to $x$:

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

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

2. Eliminate $m$ and rearrange for $a^2$:

   From (2), $m = -\frac{y}{x} \frac{dy}{dx}$. Substitute this into (1):

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

   Assuming $y \neq 0$, divide by $y$:

   $y = -\frac{1}{x} \frac{dy}{dx} (a^2 - x^2)$

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

   $xy = -a^2 y' + x^2 y'$

   $a^2 y' = x^2 y' - xy$

   $a^2 y' = x(xy' - y)$ (3)

3. Differentiate again to eliminate $a$:

   Differentiate (3) with respect to $x$:

   $a^2 \frac{d^2y}{dx^2} = \frac{d}{dx}[x(x \frac{dy}{dx} - y)]$

   $a^2 y'' = (1)(xy' - y) + x(1 \cdot y' + x \cdot y'' - y')$

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

   $a^2 y'' = xy' - y + x^2 y''$ (4)

4. Equate expressions for $a^2$ to get the differential equation:

   From (3), $a^2 = \frac{x(xy' - y)}{y'}$. …

The process of forming a differential equation from a given family of curves involves eliminating the arbitrary constants present in the equation of the family. The fundamental principle is that the order of the differential equation will be equal to the number of arbitrary constants in the original equation.

If an equation contains $n$ arbitrary constants, we must differentiate it $n$ times with respect to the independent variable (usually $x$). Each differentiation introduces a new derivative ($y', y'', \dots, y^{(n)}$). After $n$ differentiations, we will have $n+1$ equations in total: the original equation and the $n$ equations obtained by differentiation. From these $n+1$ equations, we must algebraically eliminate all $n$ arbitrary constants. The resulting equation, which contains $x, y$ and its derivatives up to order $n$, is the required differential equation.

In this problem, the given equation for the family of curves is $y^2 = m(a^2 - x^2)$. We can see that there are two arbitrary constants: $m$ and $a$. Therefore, we expect to differentiate the given equation twice to obtain a second-order differential equation.

Here's a step-by-step derivation:

1. Identify the arbitrary constants and the order of the differential equation.

   The given equation is $y^2 = m(a^2 - x^2)$.

   The arbitrary constants are $m$ and $a$. Since there are two arbitrary constants, the differential equation we form will be of the second order.

2. Differentiate the given equation once with respect to $x$.

   We apply the chain rule to $y^2$ and differentiate the right side with respect to $x$.

Using the notation $y' = \frac{dy}{dx}$:

Dividing both sides by $2$:

This equation still contains the arbitrary constant $m$. The constant $a$ is implicitly present in the original equation, but it vanished in this first differentiation because it was part of $a^2$ which is a constant.

3. Differentiate Equation 1 again with respect to $x$.

We need to differentiate $y y' = -mx$. We use the product rule on the left side and differentiate $-mx$ on the right side.

Using the notation $y'' = \frac{d^2y}{dx^2}$:

Now we have an expression for $-m$. Notice that the constant $a$ has already been eliminated, and we are left with only $m$. …