 Solution

The reduced forcing is $f(\xi )=\xi 1_{[1,\infty )}(\xi )$. Its exponential-weighted integrals converge, even though the resulting reduced function is not bounded. For $0\le x<1$, the Green's function formula givesFor $x\ge 1$ it givesUse $\int \xi \sinh \xi d\xi =\xi \cosh \xi -\sinh \xi$ and ${\int }_x^{\infty }\xi e^{-\xi }d\xi =(x+1)e^{-x}$ to simplify this to $y(x)=-x+e^{-x-1}$. Dividing by $x$ yieldsThus $a(x)=-(2/e)\sinh x/x$ with its continuous value at zero. At $x=1$, both pieces give $u=-1+e^{-2}$ and $u^{\prime }=-2e^{-2}$, so neither a Dirac delta distribution nor its derivative is introduced at the forcing jump. The solution is bounded at both ends, with $u\to -1$ at infinity. Its reduced function $y=xu\sim -x$ confirms the growth qualification in part (i).