Question

The differential equation of the family of parabolas with focus at the origin and the $X$-axis as its axis, is

• A. $-y\left(\frac{dy}{dx}\right)^2 = 2x\frac{dy}{dx} - y$
• B. $y\left(\frac{dy}{dx}\right)^2 + 2xy\frac{dy}{dx} + y = 0$
• C. $y\left(\frac{dy}{dx}\right)^2 + 4x\frac{dy}{dx} = 4xy$
• D. $y\left(\frac{dy}{dx}\right)^2 + y = 2xy\frac{dy}{dx}$

Hint:

When forming differential equations from a geometric family, the number of independent constants dictates the order of the resulting equation. Since our family depends on only one parameter ($a$), the final equation must be a first-order differential equation.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question requires us to construct the differential equation that models the family of all parabolas whose focus is located at the coordinate origin $(0,0)$ and whose principal axis of symmetry lies along the horizontal $X$-axis.

Step 2: Key Formula or Approach:
The standard Cartesian equation for a parabola with its focus at the origin and its axis along the $X$-axis is given by: $$y^2 = 4a(x + a)$$ Where $a$ is an arbitrary parameter representing the focal distance. To form the differential equation, we must differentiate this function with respect to $x$ and eliminate the parameter $a$.

Step 3: Detailed Explanation:
Differentiate both sides of the parabola equation $y^2 = 4a(x + a)$ with respect to $x$: $$2y\frac{dy}{dx} = 4a \cdot (1 + 0) \implies 2y\frac{dy}{dx} = 4a$$ Isolate the parameter $a$ from this expression: $$a = \frac{2y\frac{dy}{dx}}{4} = \frac{y}{2}\frac{dy}{dx}$$ Now substitute this expression for $a$ back into our original parabola equation to eliminate the arbitrary constant: $$y^2 = 4\left(\frac{y}{2}\frac{dy}{dx}\right) \left[ x + \frac{y}{2}\frac{dy}{dx} \right]$$ Simplify the outer leading terms: $$y^2 = \left(2y\frac{dy}{dx}\right) \left[ x + \frac{y}{2}\frac{dy}{dx} \right]$$ Divide both sides of the equation by $y$ (since $y \neq 0$): $$y = 2\frac{dy}{dx} \left[ x + \frac{y}{2}\frac{dy}{dx} \right]$$ Distribute the leading terms into the bracket expression: $$y = 2x\frac{dy}{dx} + y\left(\frac{dy}{dx}\right)^2$$ Rearrange the terms to group them on one side to match the layout in option (D): $$y\left(\frac{dy}{dx}\right)^2 + y = 2xy\frac{dy}{dx}$$

Step 4: Final Answer:
The differential equation of the family of parabolas is $y\left(\frac{dy}{dx}\right)^2 + y = 2xy\frac{dy}{dx}$, matching option (D).