Question

How do you distinguish a linear differential equation from a nonlinear one?

Short answer

Question: Determine whether the given differential equation is linear or nonlinear: \(xy' + y = x^2y^2\). Answer: The given differential equation is nonlinear, as it contains a term with the dependent variable \(y\) raised to the power of 2 (\(x^2y^2\)).

Step-by-step solution

Understand Definitions

A linear differential equation is an equation involving an unknown function and its derivatives that can be written in the form: \[\sum_{n=0}^{N} a_n(x)\frac{d^n y}{dx^n} = f(x)\] where \(a_n(x)\) are continuous functions in some common interval \(I\), \(f(x)\) is a continuous function on the interval \(I\), and \(n\) is the order of the differential equation. The nonlinear differential equation doesn't satisfy this criterion.

Inspect for Linearity

To determine if the given differential equation is linear, check if the following conditions are met: 1. The coefficients of the unknown function and its derivatives are functions of the independent variable only (no terms involving the dependent variable). 2. The dependent variable and its derivatives appear in the equation to the first power only (no squared or higher power terms, and no multiplication between the dependent variable and its derivatives).

Check for Nonlinearity

If any of the conditions mentioned in step 2 are not met, then the differential equation is nonlinear. Examples of nonlinear terms include: 1. \(y^2\) or higher powers of the dependent variable. 2. Multiplication between dependent variable and its derivatives, like \(y \frac{dy}{dx}\) or \(\frac{dy}{dx} \frac{d^2y}{dx^2}\). 3. The dependent variable or its derivatives appear in the denominator, like \(\frac{1}{y}\) or \(\frac{1}{\frac{dy}{dx}}\). After analyzing the given differential equation and comparing it with the criteria for linearity and nonlinearity, you will be able to distinguish between a linear and nonlinear differential equation.