Solving First-Order Differential Equations with Variables Separable

Understanding First-Order Differential Equations

Grasp the basics of first-order differential equations. These are equations involving derivatives of a function and the function itself.
Differentiation, at its core, is a method of finding the rate of change or slope of a line at any given point.
Familiarise yourself with the concept of separating variables, a method in which all x terms are on one side of the equation and y terms on the other side.

Understand that separation of variables is possible when an equation can be written in the form dy/dx = g(x)f(y).
Utilise algebraic manipulation to write the equation in this form.
Once the equation is in the form dy/dx = g(x)f(y), rewrite it as 1/f(y) dy = g(x) dx.

Apply the integral to both sides of the equation. For y terms use ∫1/f(y) dy and for x terms use ∫g(x) dx.
Integration is the reverse process of differentiation. It provides a method to calculate area, volume, central points, and many other useful quantities.
After integration, you will have an equation of the form Y = X + C, where C is the constant of integration determined by the initial values of the problem.

Interpret the solutions of the differential equations, recognising that these solutions can represent real life phenomena.
Notice that first-order differential equations with variables separable often appear in disciplines such as physics, engineering, economics, and biology.
Be aware that maths is not just about getting the right solution, it’s about understanding processes and making sense of phenomena that can be modelled by differential equations.

Test your knowledge by solving practical exercises. Separation of variables in differential equations requires a clear understanding of algebra, calculus, and analytical thinking.
Build your confidence by doing a variety of problems, starting with simpler ones and working your way up to complex ones.
Reinforce your understanding by solving real-world problems using these approaches. This will help you appreciate the practical uses of calculus and its application in resolving complex situations.