Let `f` be a differentiable function such that `f'(x) = f(x) + int_(0)^(2) f(x) dx and f(0) = (4-e^(2))/(3)`. Find `f(x)`.

Answer

To solve the problem, we will follow the steps outlined in the video transcript and provide a detailed explanation for each step.

### Step 1: Define the integral constant
Given the equation:
\[ f'(x) = f(x) + \int_0^2 f(x) \, dx \]
Let us denote the integral \( \int_0^2 f(x) \, dx \) as \( K \), which is a constant since the limits are fixed and \( f(x) \) is treated as a dummy variable. Thus, we can rewrite the equation as:
\[ f'(x) = f(x) + K \]

### Step 2: Differentiate both sides
Now, we differentiate both sides of the equation with respect to \( x \):
\[ f''(x) = f'(x) + 0 \]
This simplifies to:
\[ f''(x) = f'(x) \]

### Step 3: Rearranging the equation
Rearranging gives us:
\[ f''(x) - f'(x) = 0 \]

### Step 4: Solve the differential equation
This is a linear homogeneous differential equation. We can rewrite it as:
\[ \frac{f''(x)}{f'(x)} = 1 \]
Integrating both sides with respect to \( x \):
\[ \ln |f'(x)| = x + C_1 \]
where \( C_1 \) is a constant of integration. Exponentiating both sides gives:
\[ f'(x) = e^{x + C_1} = A e^x \]
where \( A = e^{C_1} \).

### Step 5: Integrate to find \( f(x) \)
Now, we integrate \( f'(x) \):
\[ f(x) = \int A e^x \, dx = A e^x + C_2 \]
where \( C_2 \) is another constant of integration.

### Step 6: Substitute back into the original equation
Now, substituting \( f(x) \) back into the original equation:
\[ f'(x) = A e^x \]
\[ f(x) = A e^x + C_2 \]
Substituting these into the equation:
\[ A e^x = (A e^x + C_2) + K \]
This simplifies to:
\[ A e^x = A e^x + C_2 + K \]
Thus, we find:
\[ 0 = C_2 + K \]
This implies:
\[ C_2 = -K \]

### Step 7: Final form of \( f(x) \)
Thus, we can express \( f(x) \) as:
\[ f(x) = A e^x - K \]

### Step 8: Use initial condition to find constants
We are given \( f(0) = \frac{4 - e^2}{3} \):
\[ f(0) = A e^0 - K = A - K \]
Setting this equal to the given value:
\[ A - K = \frac{4 - e^2}{3} \]

### Step 9: Evaluate the integral to find \( K \)
We need to evaluate \( K \):
\[ K = \int_0^2 f(x) \, dx = \int_0^2 (A e^x - K) \, dx \]
Calculating the integral:
\[ K = \left[ A e^x \right]_0^2 - 2K = A(e^2 - 1) - 2K \]
Rearranging gives:
\[ 3K = A(e^2 - 1) \]
Thus:
\[ K = \frac{A(e^2 - 1)}{3} \]

### Step 10: Substitute \( K \) back into the equation
Substituting \( K \) back into the equation \( A - K = \frac{4 - e^2}{3} \):
\[ A - \frac{A(e^2 - 1)}{3} = \frac{4 - e^2}{3} \]
Multiplying through by 3:
\[ 3A - A(e^2 - 1) = 4 - e^2 \]
This simplifies to:
\[ A(3 - e^2 + 1) = 4 - e^2 \]
Thus:
\[ A(4 - e^2) = 4 - e^2 \]
If \( 4 - e^2 \neq 0 \), then \( A = 1 \).

### Step 11: Find \( K \)
Substituting \( A = 1 \) back into the expression for \( K \):
\[ K = \frac{1(e^2 - 1)}{3} = \frac{e^2 - 1}{3} \]

### Step 12: Final function
Thus, we can write:
\[ f(x) = e^x - \frac{e^2 - 1}{3} \]
This simplifies to:
\[ f(x) = e^x + \frac{1 - e^2}{3} \]

### Final Answer
The final function is:
\[ f(x) = e^x + \frac{1 - e^2}{3} \]