= Solution
\b[The <Girsanov theorem> construction proves a <weak stochastic solution>, but does not by itself establish a <strong stochastic solution>.] It constructs $X=x+W$ first, and then constructs a new driving <Brownian motion> $B=W-\int b(X_s)ds$. Adaptation of $X$ to the original <filtration> of $W$ does not by itself prove adaptation to the smaller <natural filtration> of $B$. Inverting this relation requires an additional result; <uniqueness in law> alone is insufficient in general.
\b[For the particular coefficients in part (b), however, the literal claim that a strong solution may fail is incorrect.] The <strong existence theorem for additive-noise SDEs with bounded measurable drift> gives <strong existence> and <pathwise uniqueness> for bounded Borel state-dependent $b$ and unit diffusion. Consequently every weak realization here is determined by its driving noise and initial value. This theorem is deeper than the <Girsanov theorem> argument and no proof is needed here. Examples with path-dependent drift or with a nonconstant diffusion coefficient do not contradict it.
The valid distinction is therefore: the argument in part (b) has established weak existence and <uniqueness in law>; strong existence requires an additional theorem, and in this class that theorem is available. A primary reference for the correction is https://www.mathnet.ru/php/archive.phtml?jrnid=sm&option_lang=eng&paperid=2601&wshow=paper[On strong solutions and explicit formulas for solutions of stochastic integral equations].
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