If g(x) is continuous function in `[0, oo)` satisfying `g(1) = 1. If int_(0)^(x) 2x . g^(2)(t)dt = (int_(0)^(x) 2g(x - t)dt)^(2)`, find g(x).

Answer

To solve the problem, we need to find the function \( g(x) \) that satisfies the given integral equation:

\[ \int_0^x 2x g^2(t) \, dt = \left( \int_0^x 2g(x - t) \, dt \right)^2 \]

### Step 1: Rewrite the equation

We can rewrite the left-hand side using the property of integrals:

\[ \int_0^x 2x g^2(t) \, dt \]

The right-hand side can be rewritten by changing the variable in the integral:

\[ \int_0^x 2g(x - t) \, dt = \int_0^x 2g(t) \, dt \quad \text{(by substituting } t \text{ with } x - t\text{)} \]

Thus, our equation becomes:

\[ \int_0^x 2x g^2(t) \, dt = \left( \int_0^x 2g(t) \, dt \right)^2 \]

### Step 2: Differentiate both sides with respect to \( x \)

Differentiating both sides with respect to \( x \):

Using Leibniz's rule on the left-hand side:

\[ \frac{d}{dx} \left( \int_0^x 2x g^2(t) \, dt \right) = 2x g^2(x) + \int_0^x 2g^2(t) \, dt \]

For the right-hand side, using the chain rule:

\[ \frac{d}{dx} \left( \left( \int_0^x 2g(t) \, dt \right)^2 \right) = 2 \left( \int_0^x 2g(t) \, dt \right) \cdot 2g(x) \]

Thus, we have:

\[ 2x g^2(x) + \int_0^x 2g^2(t) \, dt = 4g(x) \int_0^x 2g(t) \, dt \]

### Step 3: Simplify the equation

Now, we can simplify the equation:

\[ 2x g^2(x) + \int_0^x 2g^2(t) \, dt = 4g(x) \int_0^x 2g(t) \, dt \]

### Step 4: Differentiate again

To isolate \( g(x) \), we differentiate both sides again with respect to \( x \):

On the left-hand side:

\[ 2g^2(x) + 2x \cdot 2g(x) g'(x) + 2g^2(x) = 4g'(x) \int_0^x 2g(t) \, dt + 4g(x) \cdot 2g(x) \]

### Step 5: Solve the resulting differential equation

We can rearrange this to find \( g'(x) \) in terms of \( g(x) \):

\[ x g'(x) = 3g(x) \]

This leads us to:

\[ \frac{g'(x)}{g(x)} = \frac{3}{x} \]

### Step 6: Integrate both sides

Integrating both sides gives:

\[ \ln g(x) = 3 \ln x + C \]

Exponentiating both sides results in:

\[ g(x) = e^C x^3 \]

### Step 7: Use the condition \( g(1) = 1 \)

Using the condition \( g(1) = 1 \):

\[ g(1) = e^C \cdot 1^3 = e^C = 1 \implies C = 0 \]

Thus, we find:

\[ g(x) = x^3 \]

### Final Answer

The function \( g(x) \) is:

\[ \boxed{x^3} \]