Solution
= Solution
Substitution gives $(\dot X^2-c^2)\phi\prime\prime-(\ddot X+\gamma\dot X)\phi\prime+\omega^2\phi=0$. Multiply by $\phi\prime$ and integrate over $\mathbb R$; decay kills the first and last <integrals>, leaving $(\ddot X+\gamma\dot X)\int(\phi\prime)^2=0$. Thus a nonconstant profile requires $X=X_\infty+Ce^{-\gamma t}$ and approaches a fixed position.