Separation Method

Calculus 2 · Axiom Academy

Split a differential equation so the y's sit on one side and the x's on the other — then integrate each side on its own. 1. Separable Means You Can Split It A differential equation is separable when its right-hand side factors into a function of x alone times a function of y alone: a function of x , times a function of y When it does, you can move each variable to its own side. Watch come apart: the y crosses left to ride with dy , the dx crosses right to ride with x . 2. Integrate Both Sides, Then Pin Down the Constant Once the variables are split, integrate each side on its own. Carry the work all the way through on with the initial condition y(0) = 1 . The traced curve below is the exact solution that comes out the other end. Integrating gives . Exponentiating turns that constant into a multiplier, with A = e^ C . The initial condition y(0) = 1 forces A = 1 , landing on the single curve y = e^ x^2/2 . 3. The Constant Chooses One Curve From a Family Before you apply an initial condition, the answer isn't one curve — it's a whole family , one curve for every value of the constant A . The animation draws several members of , then the point (0, 1) drops in and selects the only curve that passes through it. Every curve solves the equation — infinitely many, stacked vertically. Only one curve threads the point (0, 1) ; that's the particular solution y = e^ x^2/2 .

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