Question

How do differential equations with constant coefficients differ from those with variable coefficients? Give an example for each type.

Short answer

Answer: The main difference between a differential equation with constant coefficients and one with variable coefficients is that in the former, the coefficients are constants and do not change as the dependent variable changes, whereas in the latter, the coefficients are functions of the independent variable and change as the independent variable changes.

Step-by-step solution

Defining differential equations with constant coefficients

A differential equation with constant coefficients involves a function and its derivatives, with coefficients that are constants. These are easier to solve than differential equations with variable coefficients because the constant coefficients don't change as the dependent variable changes.

Defining differential equations with variable coefficients

A differential equation with variable coefficients involves a function and its derivatives, with coefficients that are functions of the independent variable. These equations are generally harder to solve because the coefficients change as the independent variable changes, often leading to more complex solutions.

Example of a differential equation with constant coefficients

Consider the second-order differential equation: \[ ay''(x) + by'(x) + cy(x) = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(y(x)\) is the function to be determined. This equation has constant coefficients as \(a\), \(b\), and \(c\) don't depend on \(x\).

Example of a differential equation with variable coefficients

Now let's consider a second-order differential equation with variable coefficients: \[ a(x)y''(x) + b(x)y'(x) + c(x)y(x) = 0 \] In this case, the coefficients \(a(x)\), \(b(x)\), and \(c(x)\) are functions of the independent variable \(x\), and hence, this is an example of a differential equation with variable coefficients.