Separation of Variables (Differential Equations) Equation, Derivation & Rearrangements

Core idea

Overview

By rearranging the differential equation so that all terms containing 'y' are with dy and all terms containing 'x' are with dx, the problem is reduced to integrating both sides independently. This technique is fundamental for solving non-linear first-order equations and finding general or particular solutions given an initial condition.

When to use: Use this method whenever a first-order differential equation can be algebraically rearranged into the product form dy/dx = f(x)g(y).

Why it matters: It is essential for modelling dynamic systems in physics, biology, and economics, such as population growth, radioactive decay, and Newton's law of cooling.

Symbols

Variables

y = Dependent Variable, x = Independent Variable, \int \frac{1}{g(y)}\,dy = \int \frac{1}{g(y)}\,dy

y

Dependent Variable

V a r iab l e

x

Independent Variable

V a r iab l e

\int \frac{1}{g ( y )} d y

\int \frac{1}{g(y)}\,dy

V a r iab l e

Walkthrough

Derivation

Derivation of Separation of Variables (Differential Equations)

This derivation demonstrates how to isolate variables in a first-order differential equation by treating the derivative as a ratio of differentials. It transforms a multivariable equation into two independent integrals.

The function g(y) is non-zero in the domain of interest.
The functions f(x) and g(y) are integrable.

1

Define the differential equation

Start with a first-order differential equation where the right-hand side is a product of a function of x and a function of y.

\frac{d y}{d x} = f (x) g (y)

Note: Ensure your equation is strictly in the product form f(x)g(y) before attempting to separate.

2

Divide by g(y)

Divide both sides by g(y) to move all terms involving y to the left-hand side.

\frac{1}{g ( y )} \frac{d y}{d x} = f (x)

Note: Be careful when dividing by g(y); check for cases where g(y) = 0 as these might represent equilibrium solutions.

3

Integrate with respect to x

Integrate both sides of the equation with respect to x.

\int \frac{1}{g ( y )} \frac{d y}{d x} d x = \int f (x) d x

Note: Include an arbitrary constant of integration (+C) on the side of the independent variable.

4

Apply change of variables

Using the substitution rule for integration, notice that dy = (dy/dx) dx, which allows the left side to be expressed strictly in terms of y.

\int \frac{1}{g ( y )} d y = \int f (x) d x

Note: This is often simplified to 'moving the dx' to the right side, but technically relies on the chain rule/substitution.

Result

\int \frac{1}{g ( y )} d y = \int f (x) d x

Source: A-Level Mathematics (OCR/Edexcel) Specification - Differential Equations

Visual intuition

Graph

Graph unavailable for this formula.

Contains advanced operator notation (integrals/sums/limits)

Why it behaves this way

Intuition

Imagine a river branching into two separate, independent streams. The rate of change of a system depends on a product of 'x-driven' growth and 'y-driven' constraints. By dividing by the y-term, we isolate the 'y-history' on one side and the 'x-history' on the other, allowing us to find the total accumulation (integral) for each dimension independently.

dy/dx

Derivative or Rate of Change

The instantaneous slope or 'growth factor' of a function at any point (x, y).

g(y)

Y-dependent constraint

The portion of the rate of change governed strictly by the current state of the dependent variable (e.g., how the population size affects its own growth rate).

f(x)

X-dependent driver

The portion of the rate of change governed by external or independent factors (e.g., time or position).

Signs and relationships

The equality symbol (=): Represents the balance between the accumulated influence of the y-component and the accumulated influence of the x-component.
Integral sign (∫): Indicates the summation or 'totaling up' of infinitesimal changes over a domain to find the underlying functional relationship.

One free problem

Practice Problem

Solve the differential equation dy/dx = 2xy, given that y=3 when x=0.

Independent Variable0

Solve for: $y$

Hint: Divide by y and multiply by dx, then integrate both sides.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating the cooling rate of a cup of coffee based on the temperature difference between the coffee and the ambient room air.

Study smarter

Tips

Always add the constant of integration 'C' immediately after performing the indefinite integration.
Check if you need to solve for y explicitly; sometimes leaving the answer in implicit form is sufficient unless specified otherwise.
Watch out for values of y that make g(y) = 0, as these represent equilibrium solutions that might be missed during division.

Avoid these traps

Common Mistakes

Forgetting to add the constant of integration (+C).
Failing to correctly rearrange the terms algebraically before integrating.
Incorrectly treating the constant of integration when using initial conditions to find a particular solution.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation demonstrates how to isolate variables in a first-order differential equation by treating the derivative as a ratio of differentials. It transforms a multivariable equation into two independent integrals.

Use this method whenever a first-order differential equation can be algebraically rearranged into the product form dy/dx = f(x)g(y).

It is essential for modelling dynamic systems in physics, biology, and economics, such as population growth, radioactive decay, and Newton's law of cooling.

Forgetting to add the constant of integration (+C). Failing to correctly rearrange the terms algebraically before integrating. Incorrectly treating the constant of integration when using initial conditions to find a particular solution.