MathematicsDifferential EquationsA-Level
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Separation of Variables (Method) Calculator

Combine integrated sides for a separable first-order ODE.

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Integration Constant

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Overview

Separation of variables is a technique used to solve first-order ordinary differential equations by isolating terms involving different variables on opposite sides of the equation. Once the variables are separated into functions of y and x, the solution is obtained by integrating both sides with respect to their respective variables.

Symbols

Variables

I_y = Left Integral Value, I_x = Right Integral Value, C = Integration Constant

Left Integral Value
Right Integral Value
Integration Constant

Apply it well

When To Use

When to use: Apply this method when a differential equation can be factored such that the derivative dy/dx equals the product of a function of x and a function of y. It is the standard approach for solving autonomous equations and basic growth or decay problems where the variables do not remain coupled.

Why it matters: This method serves as the gateway to modeling dynamic systems, allowing for the derivation of laws governing radioactive decay, fluid flow, and financial interest. It is a fundamental building block for understanding more advanced techniques in both ordinary and partial differential equations.

Avoid these traps

Common Mistakes

  • Forgetting to divide by g(y).
  • Dropping the integration constant.

One free problem

Practice Problem

A specific differential equation is integrated such that the y-integral (Iy) evaluates to 5.5 and the x-integral (Ix) evaluates to 2.1. Determine the value of the integration constant C.

Left Integral Value5.5
Right Integral Value2.1

Solve for:

Hint: Rearrange the formula to isolate C by subtracting Ix from Iy.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima
  2. Calculus: Early Transcendentals by James Stewart
  3. Wikipedia: Separation of variables
  4. A First Course in Differential Equations with Modeling Applications by Dennis G. Zill and Michael R. Cullen
  5. Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima, 11th Edition
  6. Calculus: Early Transcendentals by James Stewart, 8th Edition
  7. Separation of variables (differential equations) Wikipedia article
  8. A-Level Mathematics — First Order Differential Equations