= Solution
Both coefficient functions are linear, hence globally Lipschitz. Their norms are bounded by a constant times $|(x,y)|$, so they satisfy the <linear growth condition for an SDE>. The <existence and pathwise uniqueness theorem for a stochastic differential equation> gives a unique global <strong solution of a stochastic differential equation> for every deterministic starting point $(x_0,y_0)\in\mathbb R^2$. Linear growth prevents finite-time explosion.
The theorem gives <pathwise uniqueness>, and strong existence with <pathwise uniqueness> gives <uniqueness in law>. Thus \b[every initial value is permitted, and both forms of uniqueness hold]. The coefficients need not be bounded for this theorem; their global Lipschitz and linear-growth properties are sufficient.
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