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The characteristic polynomial after multiplication by four is $r^2+4r+4+{\pi }^2$, with roots $-2\pm i\pi$. Thus the general real solution of this constant-coefficient ordinary differential equation isEquivalently the change of dependent variable $y=e^{-2x}v$ gives $Ly=\frac{1}{4}{e}^{-2x}(v^{\prime \prime }+{\pi }^{2}v)$.