Problem 5 - Calculus Volume 3

Question

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. $$ y^{\prime \prime}+(\sin x) y^{\prime}-x y=4 y $$

Step-by-Step Solution

Answer

The equation is linear and homogeneous.

Step 1: Identify the Type of Equation

First, identify whether the given equation is linear or nonlinear. For an equation to be linear in terms of the unknown function \(y\), it should not contain any powers or functions of \(y\) or its derivatives that are other than the first power. The given equation is \(y'' + (\sin x)y' - xy = 4y\). Here, \(y''\), \(y'\), and \(y\) all appear to the first power, which suggests that the equation is linear.

Step 2: Check Linear Equation Criteria

Confirm the linearity by checking if the coefficients of \(y\), \(y'\), and \(y''\) depend only on the independent variable \(x\), not on \(y\) or its derivatives. Indeed, the coefficients are \(\sin x\) for \(y'\), \(-x\) for \(y\), and the implicit coefficient of 1 for \(y''\). This confirms it is a linear equation.

Step 3: Determine Homogeneity

To find out if the linear equation is homogeneous, check if it can be written in the form \(a_0(x)y'' + a_1(x)y' + a_2(x)y = 0\). The given equation is \(y'' + (\sin x)y' - xy = 4y\), which can be rewritten as \(y'' + (\sin x)y' - xy - 4y = 0\). Because the right-hand side after rearranging is zero, the equation is homogeneous.

Key Concepts

Homogeneous Equations

A homogeneous linear differential equation has all its terms dependent on the solution function and its derivatives, but importantly, without any standalone constant or external function. Simply put, the equation can be written in the standard form:
\[ a_0(x)y'' + a_1(x)y' + a_2(x)y = 0 \]This implies that the sum of all terms equals zero, allowing the equation to showcase interrelationships purely among the derivatives of the function.
Thus, the function balances, embodying the concept of homogeneity. This characteristic facilitates seeking solutions that take specific recursive or pattern-based forms. In our example, after rearranging the terms:
\[ y'' + (\sin x)y' - xy - 4y = 0 \]we can see that it meets the criteria for a homogeneous equation since the right-hand side simplifies to zero.

Nonlinear Equations

Nonlinear differential equations represent a different breed of complexity in mathematics. Unlike linear equations, they include terms that can be powers or products of the unknown function or its derivatives. Such behavior usually complicates finding solutions, since nonlinearity doesn't support superposition principles, which are pivotal in solving linear equations.
For instance, if any term in the equation involved \(y^2\) or \((y')^2\), it would indicate nonlinearity. No terms in our equation, \(y'' + (\sin x)y' - xy = 4y\), feature such powers or anything heavier than first-degree functions and derivatives, confirming that our equation is linear, not nonlinear.

Differential Equation Classification

Differential equations are classified primarily into linear and nonlinear categories based on how their terms are structured. A linear differential equation, like our example, contains the unknown function and its derivatives to the power of one, with coefficients independent of these unknowns—usually just functions of the independent variable.
Meanwhile, nonlinear equations could involve higher powers and products of derivatives or the function itself, introducing complexity in behavior and solution methods.

• A second classification concerns homogeneity: A linear equation is homogeneous if it can be equated to zero on the right-hand side.
• Otherwise, it is nonhomogeneous when a non-zero term exists separately, requiring approaches like utilizing particular solutions.

Understanding these classifications aids in selecting appropriate methods for tackling and solving these equations efficiently.