Separation of Variables - Solving Differential Equations

Calculus 1 · Axiom Academy

LESSON Separation of Variables When a derivative factors into an x -part times a y -part, you can split the equation in two and integrate each side on its own. 1. Recognizing a Separable Equation An equation is separable when the right-hand side splits into a product — a function of x alone times a function of y alone . If you can factor it that way, the variables can be pulled apart. f(x) uses only x • g(y) uses only y is a sum — it cannot be written as . 2. Separate, Then Integrate Both Sides Treat as a ratio of differentials you may rearrange. Multiply both sides by dx and divide by g(y) until every y (with its dy ) sits on the left and every x (with its dx ) sits on the right. Then integrate each side. 3. A Worked Example, and Pinning Down C Solve . Separating gives ; integrating gives ; exponentiating gives the general solution below — a whole family of curves, one for each value of the constant A . Never trust a solution until you've put it back into the original equation. Differentiate your answer to get , substitute both y and into the differential equation, and confirm the two sides are identical. Dropping the constant C , an algebra slip when solving for y , or forgetting needs absolute value. Variables are fully separated, both sides integrated, C fixed by the initial condition, and the answer verified by substitution. You can now spot a separable equation, split it, integrate each side, fix the constant with an initial condition, and verify the result.

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