= Solution
Fix any $t_0\geq0$ and compare $y=R$ with $y=0$. The quotient required in a global <Lipschitz condition> in the dependent variable is
$$
\frac{|f(t_0,R)-f(t_0,0)|}{|R-0|}=|R-a(t_0)|\longrightarrow\infty
\qquad(R\longrightarrow\infty).
$$
Consequently,
$$
\boxed{f\text{ is not globally Lipschitz in }y.}
$$
It is nevertheless <locally Lipschitz> in $y$: for $|y|,|v|\leq R$,
$$
|f(t,y)-f(t,v)|=|y-v|\,|y+v-a(t)|\leq(2R+C_2)|y-v|.
$$
Thus the <Picard-Lindelöf theorem> gives local existence and uniqueness for the <initial value problem>; the remaining work establishes existence on the entire half-line for the specified small data.
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