= Separable differential equation
{title2=$y'=a(t)b(y)$}
{wiki}
A first-order <ordinary differential equation> is separable if its explicit form factors as $y'=a(t)b(y)$. On an <interval> where $b(y)\ne0$, write $dy/b(y)=a(t)\,dt$ and take <antiderivatives> to obtain an <implicit solution>:
$$
\int\frac{dy}{b(y)}=\int a(t)\,dt+C.
$$
An <initial condition> determines $C$ and selects a branch and domain. The useful simplification is that the two <integrals> each involve only one variable, even when the original equation is nonlinear.
For <Newton's law of cooling>, division by $y-T_a$ gives $dy/(y-T_a)=-k\,dt$. To keep the <logarithm> dimensionless, integrate relative to the initial excess <temperature>:
$$
\log\left|\frac{y(t)-T_a}{y_0-T_a}\right|=-k(t-t_0).
$$
When $y_0\ne T_a$, <continuity> preserves the sign of the excess <temperature> until any potential zero, giving $y-T_a=(y_0-T_a)e^{-k(t-t_0)}$. This formula never reaches zero at finite time. Taking its <derivative> verifies the <ordinary differential equation>, and substitution at $t_0$ verifies the <initial condition>.
Always check what division removed. Each zero $b(y_*)=0$ gives a constant <solution of a differential equation> $y=y_*$. Cooling therefore also has the <equilibrium point> $y=T_a$, omitted by division but included in the final explicit formula. For the nonunique <initial value problem> $y'=\sqrt{|y|}$, division by $\sqrt{|y|}$ only describes the moving pieces: a valid solution may remain zero for an arbitrary time before joining one. Separation is a method for finding solutions, not a uniqueness theorem.
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