Separation of Variables Formula

Separation of variables are a method for solving first-order DEs of the form dy/dx = f(x) · g(y): rearrange to dy/(g(y)) = f(x) dx, then integrate both sides.

The Formula

dydx=f(x)g(y)⇒∫dyg(y)=∫f(x) dx+C

When to use: If the rate of change factors into a piece that depends only on x and a piece that depends only on y, you can sort them onto opposite sides of the equation—all the y-stuff on the left, all the x-stuff on the right—then integrate each side in its own variable.

Quick Example

Solve dydx=xy.
Separate: dyy=x dx.
Integrate: ln⁡∣y∣=x22+C.
Solve: y=Aex2/2 (where A=±eC).

Notation

dyg(y)=f(x) dx — all y-terms on the left with dy, all x-terms on the right with dx. +C appears on one side only.

What This Formula Means

A method for solving first-order DEs of the form dydx=f(x)⋅g(y): rearrange to dyg(y)=f(x) dx, then integrate both sides.

If the rate of change factors into a piece that depends only on x and a piece that depends only on y, you can sort them onto opposite sides of the equation—all the y-stuff on the left, all the x-stuff on the right—then integrate each side in its own variable.

Formal View

If dydx=f(x)⋅g(y) and g(y)≠0, then ∫1g(y) dy=∫f(x) dx+C. Let G(y)=∫1g(y) dy and F(x)=∫f(x) dx; then G(y)=F(x)+C defines y implicitly. Equilibrium solutions: g(y0)=0  ⟹  y=y0 is a constant solution.

Worked Examples

Example 1

easy
Solve dy/dx=xy with y(0)=2.

Answer

y=2ex2/2

First step

1
Separate: dy/y=x dx. Integrate: ln⁡∣y∣=x2/2+C.

Full solution

  1. 2
    y=Aex2/2. y(0)=2⇒A=2.
  2. 3
    Solution: y=2ex2/2.
Separate, integrate both sides, exponentiate, apply IC.

Example 2

hard
Solve the logistic DE dy/dt=y(1−y) with y(0)=1/2.

Example 3

easy
Solve dydx=2x with y(0)=5.

Common Mistakes

  • Trying to separate a non-product right side - dydx=x+y won't separate; check it factors as f(x)g(y) first.
  • Forgetting the constant of integration - add +C after integrating (on one side), then use any initial condition to find it.
  • Mishandling the dy/dx - move dx to the right and divide by g(y) properly; don't drop the differentials.

Why This Formula Matters

It is the first general technique for actually SOLVING a DE in closed form, and it solves the workhorse models — exponential growth/decay dydt=ky, logistic growth, Newton's cooling. Recognizing the separable FORM is the deciding step; if the variables won't separate, you need a different method. Recognizing it by "Can I rewrite the DE so one side has only y and dy, the other only x and dx?" — rather than by familiar numbers — is what lets a student tell it apart from slope fields and integrating factor method and plain antiderivative in a mixed problem set.

Frequently Asked Questions

What is the Separation of Variables formula?

A method for solving first-order DEs of the form dydx=f(x)⋅g(y): rearrange to dyg(y)=f(x) dx, then integrate both sides.

How do you use the Separation of Variables formula?

If the rate of change factors into a piece that depends only on x and a piece that depends only on y, you can sort them onto opposite sides of the equation—all the y-stuff on the left, all the x-stuff on the right—then integrate each side in its own variable.

What do the symbols mean in the Separation of Variables formula?

dyg(y)=f(x) dx — all y-terms on the left with dy, all x-terms on the right with dx. +C appears on one side only.

Why is the Separation of Variables formula important in Math?

It is the first general technique for actually SOLVING a DE in closed form, and it solves the workhorse models — exponential growth/decay dydt=ky, logistic growth, Newton's cooling. Recognizing the separable FORM is the deciding step; if the variables won't separate, you need a different method. Recognizing it by "Can I rewrite the DE so one side has only y and dy, the other only x and dx?" — rather than by familiar numbers — is what lets a student tell it apart from slope fields and integrating factor method and plain antiderivative in a mixed problem set.

What do students get wrong about Separation of Variables?

The procedure for separation of variables is the easy part; the trap is trying to separate a non-product right side. Asking "Can I rewrite the DE so one side has only y and dy, the other only x and dx?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Separation of Variables formula?

Before studying the Separation of Variables formula, you should understand: differential equations intro, integral.

Want the Full Guide?

This formula is covered in depth in our complete guide:

How to Integrate Rational Functions: Long Division and Partial Fractions →
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