Ordinary Differential Equations 6 | Separation of Variables

Separation of variables provides a direct route to solving certain non-autonomous ordinary differential equations by rewriting them so the time variable and the state variable appear in separate factors. For an ODE of the form ẋ = W(t, x), the method works when W can be expressed as a product G(t)·h(x). In that case, the differential equation can be rearranged into h(x) in one side and t-terms on the other, enabling integration on both sides and producing an implicit solution that can be inverted to obtain x(t). This matters because many practical models—where the dynamics change with time—still become solvable once the right algebraic structure is recognized.

The procedure begins with an initial value problem: choose a starting time t0 and require the solution satisfy x(t0) = x0. The method assumes h(x0) ≠ 0 so division by h(x) is valid in a neighborhood of x0; otherwise a constant solution can appear immediately. Once the equation is separable, the rearrangement leads to an integral form. Integrating from t0 to an arbitrary time T on the time side and from x0 to the evolving solution value α(t) on the state side yields an equation involving antiderivatives. Because antiderivatives differ by additive constants, the constants from both sides can be merged into a single constant C. The final step is to apply the inverse of the antiderivative relationship to isolate α(t), giving the solution to the initial value problem.

A worked example shows the mechanics. Consider ẋ = T^3·x^2 with an initial condition x(0) = x0. Separating variables turns the equation into (1/x^2) dx = T^3 dT. After integrating, the result involves a logarithm on the left in the simplified form used in the walkthrough and a polynomial-in-T term on the right, plus an integration constant. Exponentiating (the inverse of the natural logarithm) produces an explicit expression for the solution α(T). The constant is then fixed by enforcing the initial condition, leading to a closed-form solution.

A second example demonstrates the same workflow even when the right-hand side is more intricate: ẋ = sin(T)·e^x. Separation gives dx/e^x = sin(T) dT. Integrating produces expressions involving e^{-x} on the left and cos(T) on the right, again with an added constant. Solving for x(T) requires inverting the resulting relationship, which introduces a logarithm. The constant is determined by plugging in T = 0 and using cos(0) = 1, yielding a final solution that satisfies x(0) = x0. Across both examples, the key takeaway is that the method hinges less on memorizing a formula and more on executing the same separation, integration, inversion, and constant-matching steps reliably.