The differential equation satisfied by the system of parabolas y² = 4a(x + a) is: (1) y(dy/dx)² - 2x

Q: Wait, why do we only differentiate once? Can't we differentiate twice to get rid of $a$ faster?

That is a classic trap! The order of the differential equation is strictly tied to the number of arbitrary constants. If you differentiate twice, you are describing a more complex system than the one given. Stick to the rule: one constant, one derivative.

Q: Is it safe to divide by $y$ in the final steps? What if $y=0$?

Great observation! Mathematically, we assume $y \neq 0$ for the general solution. If $y=0$, the parabola collapses to a point or a line, which is a trivial case. For the general family of parabolas, $y \neq 0$ is a valid assumption.

Q: Why expand the equation before differentiating? Can't I just use the product rule on $4a(x+a)$?

You absolutely could! However, expanding it first to $y^2 = 4ax + 4a^2$ makes the differentiation much cleaner and reduces the chance of a sign error. In JEE, speed and accuracy are best friends—choose the path of least resistance.

Question

The differential equation satisfied by the system of parabolas $y^2 = 4a(x + a)$ is:

Options:

(A) $y\left(\frac{dy}{dx}\right)^2 - 2x\left(\frac{dy}{dx}\right) - y = 0$

(B) $y\left(\frac{dy}{dx}\right)^2 - 2x\left(\frac{dy}{dx}\right) + y = 0$

(C) $y\left(\frac{dy}{dx}\right)^2 + 2x\left(\frac{dy}{dx}\right) - y = 0$

(D) $y\left(\frac{dy}{dx}\right) + 2x\left(\frac{dy}{dx}\right) - y = 0$

Sigma Percentile · Accepted Answer

### The Geometry of Families: Unveiling the Hidden Law

Welcome, fellow traveler of the mathematical landscape. Today, we are not just solving an equation; we are uncovering the hidden law that governs a beautiful family of curves.

Imagine a collection of parabolas, all shifting and stretching along the $x$-axis, defined by the equation:

$$y^2 = 4a(x + a)$$

Each value of $a$ gives us a unique member of this family. Our goal is to find the universal differential equation that binds them all together.

### Phase 1: The Strategy of Elimination

Before we touch our pens to paper, we must understand the philosophy of the problem. We have one arbitrary constant, $a$.

In the realm of differential equations, the number of arbitrary constants dictates the order of the equation. Because we have exactly one constant, we must differentiate exactly once.

Our mission is clear: differentiate, isolate $a$, and eliminate it. It is a game of substitution, and we are the masters of the board.

### Phase 2: The Calculus of Motion

Let us begin by making our lives easier. Instead of wrestling with the product rule on the right-hand side, let us expand the equation:

$$y^2 = 4ax + 4a^2$$

Now, we apply the derivative operator $\frac{d}{dx}$ to both sides. On the left, the derivative of $y^2$ with respect to $x$ is $2y \frac{dy}{dx}$, thanks to the chain rule.

On the right, the derivative of $4ax$ is simply $4a$, and the derivative of the constant $4a^2$ is zero. We are left with a beautiful, simple relationship:

$$2y \frac{dy}{dx} = 4a$$

This is the bridge between our curve and its differential nature.

### Phase 3: The Algebraic Dance

Now, we isolate $a$. Dividing both sides by $4$, we find:

$$a = \frac{y}{2} \frac{dy}{dx}$$

This is our key. We take this expression and substitute it back into our original equation, $y^2 = 4a(x + a)$.

Substituting $a$, we get:

$$y^2 = 4 \left( \frac{y}{2} \frac{dy}{dx} ight) \left[ x + \left( \frac{y}{2} \frac{dy}{dx} ight) ight]$$

Simplifying the coefficient, $4 	imes \frac{y}{2}$ becomes $2y$. Our equation transforms into:

$$y^2 = 2y \frac{dy}{dx} \left( x + \frac{y}{2} \frac{dy}{dx} ight)$$

Now, we distribute the $2y \frac{dy}{dx}$ into the bracket. The first term becomes $2xy \frac{dy}{dx}$, and the second term becomes $y^2 \left( \frac{dy}{dx} ight)^2$.

### The Final Victory

We have the following expression:

$$y^2 = 2xy \frac{dy}{dx} + y^2 \left( \frac{dy}{dx} ight)^2$$

Since we are dealing with a family of parabolas where $y 
eq 0$, we can safely divide the entire equation by $y$. This leaves us with:

$$y = 2x \frac{dy}{dx} + y \left( \frac{dy}{dx} ight)^2$$

Rearranging the terms to the standard form, we arrive at the final governing law:

$$\mathbf{y \left( \frac{dy}{dx} ight)^2 + 2x \frac{dy}{dx} - y = 0}$$

You have successfully navigated the algebra and emerged victorious. Remember, every complex problem is just a series of simple steps waiting to be connected.

Animated Solution for Mathematics - Differential Equations: The differential equation satisfied by the system of parabolas $y^{2} = 4 a (x + a)$ is:

Select Answer:

Visualized Solution

Visualizing the Family of Curves

The Strategy

Expanding the Equation

Differentiating with respect to $x$

Applying the Chain Rule

Isolating the Constant $a$

Substituting $a$ back

Simplifying the Multiplier

Expanding the Terms

Dividing by $y$

Final Rearrangement

The Sigma Insight: Formation of Differential Equations

The Geometry of Families

Phase 1

Phase 2

Phase 3

The Final Victory

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Animated Solution for Mathematics - Differential Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The differential equation satisfied by the system of parabolas y2=4a(x+a) is: