= Solution
The <distributional derivative of the Heaviside step function> gives $H'=\delta_0$, while $((x-1)H(x-1))''=\delta_1$. A particular solution is therefore $H(x)-2(x-1)H(x-1)$.
The difference from any other solution has second derivative zero. Applying <a distribution with zero derivative is constant> twice shows that difference is affine. Thus
$$
\boxed{u=H(x)-2(x-1)H(x-1)+ax+b,\qquad a,b\in\mathbb C.}
$$
The ramp function is continuous at $x=1$, and its first derivative jumps there, producing the indicated delta source in its second derivative.
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