Animated Solution for Mathematics - Differential Equations: The differential equation of the family of curves, $x^2=4b(y+b)$, $b\in R$, is:

Imagine you are standing before a vast canvas of parabolas. Each one is unique, yet they all share a common DNA, governed by the equation:

Here, $b$ is not just a number; it is a parameter, a dial you can turn to generate an infinite family of curves. Our mission is to find the differential equation that acts as the 'master rule' for this entire family.

In the realm of differential equations, the order of the equation—the highest derivative present—is dictated by the number of arbitrary constants. Since we have only one parameter, $b$, we know with absolute certainty that we only need to differentiate once. We are looking for a first-order differential equation.

Let us begin the dance. We take our equation $x^2=4b(y+b)$ and differentiate both sides with respect to $x$.

The left side is straightforward: the derivative of $x^2$ is $2x$. On the right side, $4b$ is a constant multiplier, and the derivative of $(y+b)$ is simply $\frac{dy}{dx}$, because the derivative of the constant $b$ is zero.

So, we arrive at:

To keep our notation elegant, let us define $y^{\prime }=\frac{dy}{dx}$. Thus, our equation becomes $2x=4b{y}^{\prime }$. This is our bridge between the geometry of the parabolas and the calculus of their slopes.

Now, we face the challenge: the parameter $b$ is still lurking in our equation. To find the differential equation of the family, we must eliminate it.

We isolate $b$ from our derived equation:

This is the key that unlocks the door. We now take this expression for $b$ and substitute it back into our original equation, $x^2=4b(y+b)$.

By replacing every instance of $b$ with $\frac{x}{2y^{\prime }}$, we get:

We are not just doing algebra; we are weaving the parameter out of existence.

Now, let us simplify this expression. The term outside the bracket, $4\cdot \frac{x}{2y^{\prime }}$, simplifies beautifully to $\frac{2x}{y^{\prime }}$.

Distributing this into the parentheses, we get:

This simplifies to:

To clear the fractions, we multiply the entire equation by $(y^{\prime }{)}^2$, yielding:

Finally, assuming $x eq 0$, we divide by $x$ to reach the elegant final form:

You have just derived the governing law for an entire family of curves. Take a moment to appreciate that—you have captured an infinite number of parabolas in a single, concise mathematical statement.