What is the Dependent Variable in a Differential Equation?

The dependent variable in a differential equation is the variable whose rate of change with respect to one or more independent variables is being described by the equation. It is the variable we are trying to solve for.

In simpler terms, the dependent variable is the "output" of the differential equation; its value depends on the values of the independent variable(s) and the differential equation itself.

Explanation

Think of a function like y = f(x). Here:

• x is the independent variable (the "input").
• y is the dependent variable (the "output" that depends on the value of x).

In a differential equation, we usually have an unknown function (our dependent variable) and its derivatives. The equation relates these derivatives to the independent variable.

Examples

Here are a couple of examples to illustrate this:

• Example 1: dy/dx = 2x

  • In this differential equation, y is the dependent variable (it's a function of x whose derivative is given). x is the independent variable. We're trying to find the function y(x) that satisfies this equation.

• Example 2: d^2s/dt^2 = -g (representing acceleration due to gravity)

  • Here, s (position) is the dependent variable, t (time) is the independent variable, and g (acceleration due to gravity) is a constant. The differential equation describes how the position s changes with time t.

Key Takeaways

• The dependent variable is the unknown function you are solving for in a differential equation.
• Its value depends on the independent variable(s) and the differential equation.
• The differential equation describes the relationship between the dependent variable, its derivatives, and the independent variable(s).