Question

Can a differential equation involve more than one independent variable? Can it involve more than one dependent variable? Give examples.

Short answer

Answer: Yes, a differential equation can involve more than one dependent variable, as shown in the example of a system of 2 ordinary differential equations: $\frac{d{y}_1}{dt}=y_1^2+y_2^2$ and $\frac{d{y}_2}{dt}=y_1+y_2$, where $y_1=y_1(t)$ and $y_2=y_2(t)$ are both dependent variables and t is the independent variable. Moreover, a differential equation can also involve more than one independent variable. This type of equation is known as a partial differential equation (PDE). An example is the 2D heat equation, which is a PDE with two independent variables x and y: $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$, where u = u(x, y) is the dependent variable and x, y are independent variables.

Step-by-step solution

Dependent variables in differential equations

A differential equation can indeed involve more than one dependent variable. For example, consider a system of 2 ordinary differential equations: $\frac{d{y}_1}{dt}=y_1^2+y_2^2$ and $\frac{d{y}_2}{dt}=y_1+y_2$, where $y_1=y_1(t)$ and $y_2=y_2(t)$ are both dependent variables and t is the independent variable.

Independent variables in differential equations

A partial differential equation (PDE) is an equation involving multiple independent variables and their partial derivatives. For example, a 2D heat equation is a PDE having two independent variables x and y: $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$, where u = u(x, y) is the dependent variable and x, y are independent variables.