If y = y(x) satisfies the differential equation 8√x(√9+√x)dy=(√4+√9+√x)⁻¹ dx, x 0 and y(0) = √7, t

Q: Why can't we just integrate this directly? It looks like a standard power rule problem.

The complexity of the nested square roots makes direct integration impossible. We must simplify the variable of integration first. By choosing a clever substitution, we transform a seemingly impossible integral into a standard one.

Q: How do I know which part of the expression to pick for the substitution $t$?

Look for the most complex part of the expression. Often, the derivative of that part will be present elsewhere in the equation, allowing for a clean substitution. In this case, the innermost structure $4+\sqrt{9+\sqrt{x}}$ is the perfect candidate.

Q: I got $C \neq 0$. Did I mess up the initial condition?

Check your arithmetic. The initial condition $y(0) = \sqrt{7}$ is crucial. Ensure you evaluate the nested roots correctly at $x=0$. Remember, $\sqrt{9} = 3$, so $4+3=7$.

Question

If $y = y(x)$ satisfies the differential equation $8\sqrt{x}(\sqrt{9+\sqrt{x}})dy=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1} dx$, $x > 0$ and $y(0) = \sqrt{7}$, then $y(256) =$

Options:

(A) 3

(B) 9

(C) 16

(D) 80

Sigma Percentile · Accepted Answer

### The Anatomy of the Beast

Welcome to the arena, fellow traveler. Today, we face a differential equation that looks like it was designed to haunt your dreams.

When you first glance at $8\sqrt{x}(\sqrt{9+\sqrt{x}})dy = (\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1} dx$, it is perfectly natural to feel a surge of panic. The nested square roots and the variables tangled in a web of radicals are a classic JEE Advanced tactic to test your composure.

But remember, every complex problem is just a collection of simple steps waiting to be revealed. Let us break this beast down.

### Phase 1: The Setup

Our first objective is to bring order to this chaos. We need to separate our variables by isolating $dy$ on one side and all $x$ terms on the other.

$$dy = \frac{1}{8\sqrt{x}\sqrt{9+\sqrt{x}}\sqrt{4+\sqrt{9+\sqrt{x}}}} dx$$

Now, look at that denominator. It is a mess, but it is a **structured mess**. The key to solving this lies in recognizing the pattern of a function nested within a function, which is a massive hint to use the method of substitution.

### Phase 2: The Art of Substitution

To proceed, we choose our substitution variable $t$ as the most complex, innermost part of the expression: $t = 4+\sqrt{9+\sqrt{x}}$. This might seem like a bold choice, but trust the process.

When we differentiate this with respect to $x$, we are essentially peeling back the layers of an onion. Using the chain rule, we find:

$$\frac{dt}{dx} = \frac{1}{2\sqrt{9+\sqrt{x}}} \cdot \frac{d}{dx}(9+\sqrt{x}) = \frac{1}{2\sqrt{9+\sqrt{x}}} \cdot \left(\frac{1}{2\sqrt{x}}\right)$$

Simplifying this, we obtain the differential form:

$$dt = \frac{1}{4\sqrt{x}\sqrt{9+\sqrt{x}}} dx$$

### Phase 3: The Collapse

Now, look back at our original equation. The $dt$ expression is almost identical to the $dx$ part of our original equation, save for a constant factor. We can rewrite the original equation as:

$$dy = \frac{1}{2} \cdot \frac{1}{\sqrt{t}} \cdot \left(\frac{1}{4\sqrt{x}\sqrt{9+\sqrt{x}}} dx\right)$$

Substituting $dt$ for the term in the parentheses, we arrive at the beautiful, elegant form:

$$dy = \frac{1}{2\sqrt{t}} dt$$

This is the moment where the tension breaks. The monster has collapsed into a simple, solvable integral.

### Phase 4: The Integration and the Constant

Integrating both sides is now trivial. The integral of $dy$ is $y$, and the integral of $\frac{1}{2\sqrt{t}} dt$ is simply $\sqrt{t} + C$.

So, our general solution is:

$$y(x) = \sqrt{4+\sqrt{9+\sqrt{x}}} + C$$

We are given the initial condition $y(0) = \sqrt{7}$. Plugging $x=0$ into our equation, we get $\sqrt{4+\sqrt{9+0}} + C = \sqrt{7}$, which simplifies to $\sqrt{7} + C = \sqrt{7}$. Thus, $C = 0$.

### Phase 5: The Final Reveal

Our particular solution is simply $y(x) = \sqrt{4+\sqrt{9+\sqrt{x}}}$. Finally, we evaluate at $x=256$:

$$y(256) = \sqrt{4+\sqrt{9+\sqrt{256}}} = \sqrt{4+\sqrt{9+16}} = \sqrt{4+\sqrt{25}} = \sqrt{4+5} = \sqrt{9} = 3$$

And there it is. The final answer is **3**. The lesson here is not just about the math; it is about the mindset. When you face a problem that looks impossible, look for the structure, find the substitution, and watch as the complexity melts away.

Animated Solution for Mathematics - Differential Equations: If $y = y (x)$ satisfies the differential equation $8 x (9 + x) d y = (4 + 9 + x)^{- 1} d x$ , $x > 0$ and $y (0) = 7$ , then $y (256) =$

Select Answer:

Visualized Solution

The Differential Equation

Choosing the Substitution $t$

Differentiating $t$

Simplifying $d t$

Simplified Differential Equation

Integrating Both Sides

General Solution

Applying Initial Condition

Solving for $C$

Finding $y (256)$

Evaluating Middle Root

Final Calculation

The Sigma Insight: Variable Separable Method

The Anatomy of the Beast

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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