` (dy)/(dx) = x sqrt(25 -x^(2))`

Answer

To solve the differential equation \( \frac{dy}{dx} = x \sqrt{25 - x^2} \), we will follow these steps:

### Step 1: Separate the Variables
We can separate the variables \( y \) and \( x \) by moving all terms involving \( y \) to one side and all terms involving \( x \) to the other side:
\[ dy = x \sqrt{25 - x^2} \, dx \]

### Step 2: Integrate Both Sides
Next, we will integrate both sides. The left side integrates to \( y \), and we will integrate the right side:
\[ \int dy = \int x \sqrt{25 - x^2} \, dx \]
This gives us:
\[ y = \int x \sqrt{25 - x^2} \, dx \]

### Step 3: Use Substitution for the Integral
To solve the integral \( \int x \sqrt{25 - x^2} \, dx \), we can use the substitution:
\[ t = 25 - x^2 \quad \Rightarrow \quad dt = -2x \, dx \quad \Rightarrow \quad dx = -\frac{dt}{2x} \]
From the substitution, we also have \( x^2 = 25 - t \), so \( x = \sqrt{25 - t} \).

### Step 4: Substitute and Simplify
Now, substituting \( t \) into the integral:
\[ \int x \sqrt{25 - x^2} \, dx = \int x \sqrt{t} \left(-\frac{dt}{2x}\right) = -\frac{1}{2} \int \sqrt{t} \, dt \]
This simplifies to:
\[ -\frac{1}{2} \cdot \frac{2}{3} t^{3/2} = -\frac{1}{3} t^{3/2} \]

### Step 5: Substitute Back
Now we substitute back \( t = 25 - x^2 \):
\[ -\frac{1}{3} (25 - x^2)^{3/2} \]

### Step 6: Write the Final Solution
Thus, we have:
\[ y = -\frac{1}{3} (25 - x^2)^{3/2} + C \]
where \( C \) is the constant of integration.

### Final Answer
The solution to the differential equation is:
\[ y = -\frac{1}{3} (25 - x^2)^{3/2} + C \]