Separable differential equation ( $y^{'} = a (t) b (y)$ )

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) \neq = 0

, write

d y / b (y) = a (t) d t

and take antiderivatives to obtain an implicit solution:

\int \frac{d y}{b ( y )} = \int a (t) d t + C .

(1)

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

d y / (y - T_{a}) = - k d t

. To keep the logarithm dimensionless, integrate relative to the initial excess temperature:

lo g \frac{y ( t ) - T _{a}}{y _{0} - T _{a}} = - k (t - t_{0}) .

(2)

When

y_{0} \neq = 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^{'} = ∣ y ∣

, division by

∣ 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.