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# Determine functions $y= 2e^{-x}+ 3e^{2x}$ is solution to which of following differential equation

$(a)\;\frac{d^2y}{dx^2}=2y+x \\ (b)\;\frac{d^2y}{dx^2}=\frac{dy}{dx}+2y \\ (c)\;\frac{d^2y}{dx^2}=\frac{dy}{dx}+x \\ (d)\;\frac{d^2y}{dx^2}=\frac{dy}{dx}+y$

$y= 2e^{-x}+3e^{2x}$
$\large\frac{dy}{dx}$$=-2e^{-x}+6 e^{2x} \large\frac{d^2y}{dx^2}$$=2 e^{-x}+12e^{2x}$
Hence b is the correct answer.