A function `f: RvecR` satisfies `sinxcosy(f(2x+2y)-f(2x-2y)=cosxsiny(f(2x+2y)+f(2x-2y))dot` If `f^(prime)(0)=1/2,t h e n` `a)f^('')(x)=f(x)=0` `b)4f^('')(x)+f(x)=0` `c)f^('')(x)+f(x)=0` `d)4f^('')(x)-f(x)=0`

Options

A. `f''(x)=f(x)=0`
B. `4f''(x)+f(x)=0`
C. `f''(x)+f(x)=0`
D. `4f''(x)-f(x)=0`

Explanation

To solve the problem, we are given the functional equation involving \( f \) and we need to find the correct differential equation that \( f \) satisfies. Let's go through the steps systematically.

### Step 1: Rewrite the functional equation
The given functional equation is:

\[ \sin x \cos y (f(2x + 2y) - f(2x - 2y)) = \cos x \sin y (f(2x + 2y) + f(2x - 2y)) \]
...