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
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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.
Rewritten readings were written for this dataset.
More to read
Separating variables to solve
An equation is separable when algebra can put it in the form dy/dx = f(x)g(y). Move the y-dependent factors with dy and the x-dependent factors with dx, then integrate both sides. Combine the integration constants into one C.
Varying C describes a family of solutions; an initial condition can determine C and select a curve. Check the result's valid interval. Dividing by a y-dependent expression may exclude constant-valued solutions or restrict where the working applies.
Separable differential equations
This article walks through separation as a practical skill: recognise the product structure, divide to isolate, then integrate. Worked examples show the standard moves, including absorbing constants into a single C and solving explicitly for y when the algebra allows.
It also warns about the classic slips: dropping the constant, dividing by zero without noting the lost solution, and forgetting that the answer needs an interval. Reading it with pencil in hand turns the method into reflex.
Checking a differential-equation solution
A function solves a differential equation when it satisfies that equation throughout an interval. Substitute the function and its derivatives to verify this. During separation, check that division and integration have not introduced invalid results or excluded solutions.
An initial condition can determine the integration constant and select a member of the solution family. State the interval where that function satisfies the equation; the formula need not remain valid beyond it.
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Optional connections
13 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.