Separable Equations
When the equation factors into a part in y and a part in x, separate the variables and integrate both sides. The constant of integration is where the initial condition enters.
What a learner can do afterwards
- Separate and integrate an equation, keeping the constant
- Apply an initial condition and state the interval where the solution is valid
- Recognise an equation that cannot be separated
1 · Read
A separable equation splits into a y-piece times an x-piece: dy/dx = f(x) times g(y), perhaps after a little algebra. You park every y-piece with dy on one side and every x-piece with dx on the other. Then you integrate each side on its own. One constant of integration ties the two halves into a whole family of curves.
Take dy/dx = 2x with y(0) = 3. Integrate: y = x squared + C. The condition fixes C: 3 = 0 + C, so C = 3. At x = 1 you get y = 1 + 3 = 4. Keep C from the start and the condition simply names it.
The condition selects one curve, and you must say where it stays valid. Dividing by y-expressions can hide solutions: dy/dx = x y is also solved by y = 0. The nonzero family is y = C e to the x squared over 2, and y(0) = 2 gives C = 2, valid for all x. With dy/dx = 2y over x for x greater than 0, you get ln|y| = 2 ln|x| + C, so y = C x squared. Then y(1) = 1 gives C = 1 and y(3) = 9, trusted only for x greater than 0 because you divided by x.
Test the shape first: products and quotients of one x-piece and one y-piece split, while sums like x + y never split. Never drop C: the condition fixes it, it does not replace it. Always name the interval where your divisions stay legal.
Separate the variables, integrate both sides, let the condition fix C, and state the interval.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.