Solving Differential Equations by Separation of Variables

Calculus 1 · Axiom Academy

EXAMPLE Separation of Variables Solve a first-order differential equation by separating the variables, integrating, and applying an initial condition. Solve the initial value problem with y(0) = 2 . Nice work — you solved a separable differential equation from start to finish. Separate, then integrate: get every y on one side and every x on the other, so each side integrates on its own. One constant is enough: the two integration constants fold into a single C , and after exponentiating, absorbs it. The initial condition pins it down: y(0) = 2 forces A = 2 , giving the unique solution y = 2e^ x^2/2 . Separation of variables turns a differential equation into two ordinary integrals — the workhorse first method for first-order equations.

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