If `y=y(x)` satisfies the differential equation `8sqrt(x)(sqrt(9+sqrt(x)))dy=(sqrt(4+sqrt(9+sqrt(x))))^(-1)dx ,x >0a n dy(0)=sqrt(7,)` then `y(256)=` 16 (b) 80 (c) 3 (d) 9

Options

A. 3
B. 9
C. 16
D. 80

Explanation

To solve the differential equation given in the problem, we will follow these steps:

### Step 1: Rewrite the differential equation
The given differential equation is:
\[ 8\sqrt{x} \sqrt{9 + \sqrt{x}} \, dy = \left(\sqrt{4 + \sqrt{9 + \sqrt{x}}}\right)^{-1} \, dx \]
We can rearrange this to isolate \(dy\):
\[ dy = \frac{dx}{8\sqrt{x} \sqrt{9 + \sqrt{x}} \sqrt{4 + \sqrt{9 + \sqrt{x}}}} \]

### Step 2: Substitution
Let’s make a substitution to simplify the expression. Set:
\[ T = 4 + \sqrt{9 + \sqrt{x}} \]
Then differentiate \(T\) with respect to \(x\):
\[ \frac{dT}{dx} = \frac{1}{2\sqrt{9 + \sqrt{x}}} \cdot \frac{1}{2\sqrt{x}} = \frac{1}{4\sqrt{x}\sqrt{9 + \sqrt{x}}} \]
Thus, we can express \(dx\) in terms of \(dT\):
\[ dx = 4\sqrt{x}\sqrt{9 + \sqrt{x}} \, dT \]

### Step 3: Substitute back into the equation
Substituting \(dx\) into our expression for \(dy\):
\[ dy = \frac{4\sqrt{x}\sqrt{9 + \sqrt{x}} \, dT}{8\sqrt{x}\sqrt{9 + \sqrt{x}} \sqrt{4 + \sqrt{9 + \sqrt{x}}}} = \frac{1}{2\sqrt{T}} \, dT \]

### Step 4: Integrate both sides
Now we can integrate both sides:
\[ \int dy = \int \frac{1}{2\sqrt{T}} \, dT \]
This gives:
\[ y = \sqrt{T} + C \]

### Step 5: Substitute back for \(T\)
Substituting back for \(T\):
\[ y = \sqrt{4 + \sqrt{9 + \sqrt{x}}} + C \]

### Step 6: Use the initial condition to find \(C\)
We know from the problem that \(y(0) = \sqrt{7}\). We need to find \(C\):
\[ \sqrt{7} = \sqrt{4 + \sqrt{9 + \sqrt{0}}} + C \]
This simplifies to:
\[ \sqrt{7} = \sqrt{4 + 3} + C \implies \sqrt{7} = \sqrt{7} + C \]
Thus, \(C = 0\).

### Step 7: Final expression for \(y\)
So, we have:
\[ y = \sqrt{4 + \sqrt{9 + \sqrt{x}}} \]

### Step 8: Calculate \(y(256)\)
Now we need to find \(y(256)\):
\[ y(256) = \sqrt{4 + \sqrt{9 + \sqrt{256}}} \]
Calculating step by step:
\[ \sqrt{256} = 16 \implies y(256) = \sqrt{4 + \sqrt{9 + 16}} = \sqrt{4 + \sqrt{25}} = \sqrt{4 + 5} = \sqrt{9} = 3 \]

### Final Answer
Thus, \(y(256) = 3\).