Ordinary Differential Equations 6 | Separation of Variables [dark version]

Separation of variables provides a practical route to solving certain non-autonomous ordinary differential equations by rewriting them so the time-dependent part and the state-dependent part sit in separate factors. For an ODE of the form ẋ = G(t)·h(x), the method works when h(x) is not zero at the initial state, letting the equation be rearranged into 1/h(x)·dx = G(t)·dt. With an initial value condition x(t0) = x0, integrating both sides from t0 to an arbitrary time T produces an implicit solution that can be expressed using antiderivatives and then inverted to obtain x(T). The key payoff is that the same calculus steps—rearrange, integrate, apply initial data, invert—repeat in a predictable pattern.

The procedure begins by checking the “nontrivial” case: if h(x0) = 0, the constant solution x(t) ≡ x0 already satisfies the equation, so there’s nothing to separate. Otherwise, near x0 the function h(x) stays nonzero, so dividing by h(x) is legitimate. After rearranging, the method leverages the fundamental theorem of calculus by integrating both sides with respect to their natural variables. Antiderivatives appear on both sides: one for 1/h(x) and one for G(t). Because antiderivatives differ only by an additive constant, the constants can be merged into a single real constant C. The final step is to invert the resulting relationship to solve explicitly for the solution function α(t) (the trajectory satisfying the initial condition).

A worked example makes the mechanics concrete. For ẋ = T^3·x^2 with x(0) = x0, separation is possible because the right-hand side is a product of a pure time factor T^3 and a pure state factor x^2. Rearranging yields dx/x^2 = T^3 dt, and integrating produces an expression involving the natural logarithm of |x| on the left and a power of t on the right, plus a constant. Exponentiating (using the inverse of the natural logarithm) and then enforcing x(0) = x0 determines the constant, giving the explicit solution in terms of an exponential factor scaled by x0.

A second example shows the method still applies when the state factor is exponential. For ẋ = sin(t)·e^x, separation leads to dx/e^x = sin(t) dt. Integrating gives an implicit relation involving e^{-x} on the left and cos(t) on the right, again with an added constant. Solving for x requires applying the inverse of the exponential relationship, producing a logarithmic expression. The constant is then fixed by the initial condition x(0) = x0, using cos(0) = 1 to simplify the algebra. The result is a fully determined solution α(t), obtained through the same repeated steps: separate, integrate, combine constants, invert, and apply initial data.

Overall, the method’s importance lies in turning a differential equation into an algebraic problem after integration. Rather than memorizing a general formula, the emphasis is on executing the separation workflow reliably on each example—especially for non-autonomous equations where time dependence would otherwise complicate direct solution.