Separation of Variables (Differential Equations) Calculator
A method for solving first-order differential equations where the variables can be separated to opposite sides of the equation.
Formula first
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.
Symbols
Variables
y = Dependent Variable, x = Independent Variable, \int \frac{1}{g(y)}\,dy = \int \frac{1}{g(y)}\,dy
Apply it well
When To Use
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.
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.
One free problem
Practice Problem
Solve the differential equation dy/dx = 2xy, given that y=3 when x=0.
Solve for:
Hint: Divide by y and multiply by dx, then integrate both sides.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Stewart, J. (2015). Calculus: Early Transcendentals.
- A-Level Mathematics: Differential Equations (OCR/Edexcel Specification)
- A-Level Mathematics (OCR/Edexcel) Specification - Differential Equations