# Separable differential equation

An equation for dy/dx that can be rearranged to have all the y terms on one side and all the x terms on the other. Then integrate both sides.

Maths (university) · Part 06: Integration techniques and uses · Card 8 of 9 · Last updated 2026-10-19

## Analogy

Sorting laundry: all the y-items into one pile, all the x-items into the other.

## Example

**dy/dx = 2xy, with y(0) = 3**

Separate: dy/y = 2x dx.
Integrate: ln y = x^2 + C.
So y = A e^(x^2).
y(0) = 3 gives A = 3, so y = 3e^(x^2).

## In practice

Models growth, cooling and decay, where the rate depends on the current amount.

## Practice

### Flashcards

- **What is the solution of dy/dx = 2xy with y(0) = 3?** y = 3e^(x^2).

## Questions people ask

### What is a differential equation?

It is an equation that links a function to its derivatives. Instead of asking for a number, you must find a whole function. For dy/dx = 2y, the solutions are y equals A times e to the 2x, where A depends on the starting condition.

### What is an initial condition for?

The general solution contains a constant, so there are infinitely many solutions. An initial condition, such as y(0) = 3, picks the one that passes through a given point. It lets you work out the value of the constant. In the example, y(0) = 3 forces A to be 3, so only one curve remains.

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