Understanding the Differential Equation Representing a Family of Curves

Introduction: In mathematical analysis, a family of curves often represents a set of related functions defined by a variable parameter. When we aim to find the differential equation that represents this family, we need to eliminate the arbitrary constants using differentiation. This article explores how to find the differential equation for the family of curves given by the equation y - b^2 4x - a.

Step-by-Step Solution

Let's start by rewriting the given equation for clarity:

Step 1: Rewrite the Equation

The initial equation is:

(y - b^2 4x - a)

Step 2: Implicit Differentiation

Next, we need to differentiate both sides of the equation with respect to (x) using implicit differentiation:

[frac{d}{dx}(y - b^2) frac{d}{dx}(4x - a)]

The left side differentiates to:

[2y - 2bfrac{db}{dx} 4]

The right side simplifies to:

[4 0]

So, the equation becomes:

[2y - 2bfrac{db}{dx} 4]

We can simplify this to:

[y - bfrac{dy}{dx} 2]

Step 3: Solve for (frac{dy}{dx})

Isolating (frac{dy}{dx}) from the equation:

[frac{dy}{dx} frac{y - 2}{b}]

Step 4: Express (b) in Terms of (y) and (x)

Recall from the original equation that:

[y - b^2 4x - a]

So, we can express (b) as:

[b pmsqrt{frac{y - 4x a}{1}}]

Substituting (b) back into the expression for (frac{dy}{dx}):

[frac{dy}{dx} frac{y - 2}{pmsqrt{y - 4x a}}]

Step 5: Eliminate (a) and (b)

To eliminate (a), we note that:

[x - a frac{y - b^2}{4}]

Substituting (b) back in:

[x - a frac{y - left(pmsqrt{y - 4x a}right)^2}{4}]

Simplifying this:

[x - a frac{y - (y - 4x a)}{4}]

Which simplifies to:

[x - a x - a]

Now, we need to square (frac{dy}{dx}) to eliminate the square root:

[left(frac{dy}{dx}right)^2 frac{(y - 2)^2}{y - 4x a}]

Since (x - a frac{y - b^2}{4}), we substitute (a 4x - y b^2). This simplifies the equation to:

[left(frac{dy}{dx}right)^2 frac{1}{4}left(y - b^2right)^2]

This is the differential equation that represents the family of curves.

Conclusion

The differential equation for the given family of curves is:

[left(frac{dy}{dx}right)^2 frac{1}{4}(y - b^2)^2]

Thus, we have derived the differential equation that describes the family of parabolas represented by the equation y - b^2 4x - a.

References

[1] Wikipedia, Differential Equations

[2] Lamar University, Differential Equations