 Solution

The repeated-root constant-coefficient differential equation has characteristic polynomial $(r+2{)}^2$. Its two fundamental solutions areThey are linearly independent: their Wronskian is $e^{-4x}$, which never vanishes. For the forced equation, exploit the repeated factor by writing $y=e^{-2x}u$. Direct differentiation gives $(D+2{)}^{2}y=e^{-2x}{u}^{\prime \prime }$, so the forcing reduces to $u^{\prime \prime }=1$. The initial data imply $u(0)=0$ and $u^{\prime }(0)=0$, hence $u=x^2/2$. ThereforeThe extra factor $x^2$ reflects resonance of the forcing with the repeated characteristic root; the exponential substitution obtains it without guessing a particular solution.