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medium difficulty8 min

Variable-Separable Differential Equations

A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side.

In one line: Separate first, integrate second; never integrate a mixed side.

Practice This Concept

Understand it

The standard idea is to rewrite dy/dx = f(x)g(y) as dy/g(y) = f(x) dx, then integrate both sides. The constant of integration is essential because the solution represents a family of curves.

Example

For dy/dx = xy, separate variables: dy/y = x dx. Integrate: ∫dy/y = ∫x dx, so log|y| = x2/2 + C.

Formula and rule

Key form: dy/dx = f(x)g(y).

Separation gives dy/g(y) = f(x) dx, provided g(y) is not zero on the interval considered.

Then ∫dy/g(y) = ∫f(x) dx + C.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Example: Solve dy/dx = x(1 + y2).

  1. 1Separate: dy/(1 + y2) = x dx.
  2. 2Integrate both sides: tan−1y = x2/2 + C.
  3. 3Hence the general solution is tan−1y - x2/2 = C.

Method you can reuse

  1. 1Rewrite the equation in differential form if useful.
  2. 2Move all y terms with dy to one side.
  3. 3Move all x terms with dx to the other side.
  4. 4Integrate both sides.
  5. 5Add the arbitrary constant.
  6. 6Simplify only if it preserves the solution correctly.

Visual tip

A diagram is not required because the solution method is symbolic. A slope-field visual may help conceptually, but CBSE solving questions usually require algebraic separation and integration.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Solve the separated form dy/y = 3x2 dx.

Think first, then show answer
Integrating gives log|y| = x3 + C, or y = Cex3 where C is a non-zero arbitrary constant; y = 0 may also satisfy the original form if allowed.

Common wrong answer

Forgetting + C after integration, or writing ∫dy/(1 + y2) as log(1 + y2) instead of tan−1y.

Don't mix this up

Students often integrate dy/dx = f(x)g(y) directly with respect to x without separating y terms correctly, which treats y like a constant.

Exam tip

After separating variables, check that one side contains only y with dy and the other contains only x with dx before integrating.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Usually asked as a direct solving problem, sometimes after rearrangement. The equation may appear as M(y)dy = N(x)dx or dy/dx = f(x)g(y).
  1. 1

    Enough for 1 mark

    A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side.

  2. 2

    Add for 2 marks

    Example: For dy/dx = xy, separate variables: dy/y = x dx. Integrate: ∫dy/y = ∫x dx, so log|y| = x2/2 + C.

  3. 3

    Add for 3 marks

    The standard idea is to rewrite dy/dx = f(x)g(y) as dy/g(y) = f(x) dx, then integrate both sides.

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