Let `f(x)` be a polynomial satisfying f(0)=2 , `f'(0)=3` and `f''(x)=f(x)` then f(4) equals

Options

A. `(5(e^(8)+1))/(2e^(4))`
B. `(5(e^(8)-1))/(2e^(4))`
C. `(2e^(4))/(5(e^(8)-1))`
D. `(2e^(4))/(5(e^(8)+1))`

Explanation

To solve the problem, we need to find the polynomial \( f(x) \) that satisfies the given conditions. Let's go through the steps systematically.

### Step 1: Set up the differential equation
We are given that \( f''(x) = f(x) \). This is a second-order linear differential equation.

### Step 2: Write the characteristic equation
The characteristic equation for \( f''(x) = f(x) \) is:
\[ r^2 - 1 = 0 \]
This gives us the roots \( r = 1 \) and \( r = -1 \).

### Step 3: Write the general solution
The general solution of the differential equation is:
\[ f(x) = A e^x + B e^{-x} \]
where \( A \) and \( B \) are constants to be determined.

### Step 4: Use initial conditions
We are given two initial conditions:
1. \( f(0) = 2 \)
2. \( f'(0) = 3 \)

#### Applying the first condition:
Substituting \( x = 0 \) into the general solution:
\[ f(0) = A e^0 + B e^0 = A + B = 2 \quad \text{(Equation 1)} \]

#### Applying the second condition:
First, we need to find \( f'(x) \):
\[ f'(x) = A e^x - B e^{-x} \]
Now substituting \( x = 0 \):
\[ f'(0) = A e^0 - B e^0 = A - B = 3 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
Now we have a system of equations:
1. \( A + B = 2 \)
2. \( A - B = 3 \)

Adding these two equations:
\[ (A + B) + (A - B) = 2 + 3 \implies 2A = 5 \implies A = \frac{5}{2} \]

Substituting \( A \) back into Equation 1:
\[ \frac{5}{2} + B = 2 \implies B = 2 - \frac{5}{2} = -\frac{1}{2} \]

### Step 6: Write the specific solution
Now we have \( A \) and \( B \):
\[ f(x) = \frac{5}{2} e^x - \frac{1}{2} e^{-x} \]

### Step 7: Calculate \( f(4) \)
Now we need to find \( f(4) \):
\[ f(4) = \frac{5}{2} e^4 - \frac{1}{2} e^{-4} \]

### Step 8: Simplify the expression
This can be simplified as:
\[ f(4) = \frac{5}{2} e^4 - \frac{1}{2 e^4} \]

### Final Answer
Thus, the value of \( f(4) \) is:
\[ f(4) = \frac{5 e^4 - 1}{2 e^4} \]