# Which of the following differential equations has $y\;=\;c_1e^x+c_2e^{-x}$ as the general solution?

$(A)\;\frac{d^2y}{dx^2}+y=0\quad(B)\;\frac{d^2y}{dx^2}-y=0\quad(C)\;\frac{d^2y}{dx^2}+1=0\quad(D)\;\frac{d^2y}{dx^2}-1=0$

Toolbox:
• Differentiation of $e^x = e^x$
Step 1:
Given: $y = c_1 e^x + c_2e^{-x}$
Differentiating this we get
$c_1e^x - c_2e{-x} = y'$
Step 2:
Again differentiating we get,
$c_1e^x + c_2e^{-x} = y''$
$y = y''$
Hence $y'' - y = 0$
Hence option B is the correct answer.