Separable differential equation

Definition

A separable differential equation is a type of ordinary differential equation that can be written as the product of a function of the independent variable and a function of the dependent variable. Such equations can be solved by separating the variables and integrating both sides.

5 Must Know Facts For Your Next Test

1. The general form of a separable differential equation is $\frac{dy}{dx} = g(x)h(y)$.
2. To solve, rewrite it as $\frac{1}{h(y)} dy = g(x) dx$ and then integrate both sides.
3. Always include the constant of integration when integrating both sides.
4. The solution often involves implicit functions that may require further algebraic manipulation to solve explicitly for $y$.
5. Separable differential equations are typically among the first types of differential equations taught because they are simpler to solve.

Review Questions

• What is the general form of a separable differential equation?
• How do you solve a separable differential equation?
• Why is it important to include the constant of integration when solving separable differential equations?

Related terms

Ordinary Differential Equation (ODE): An equation involving derivatives with respect to only one variable.

Integration: The process of finding an integral or antiderivative of a function.

Constant of Integration: An arbitrary constant added to an indefinite integral, representing all possible antiderivatives.