Problem 5 - Calculus 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 nonlinear due to the presence of the term $4y$.

Step 1: Identify the Order of the Equation

The given equation is \( y'' + (\sin x) y' - x y = 4 y \). The highest derivative present is \( y'' \), which is the second derivative, so this is a second-order equation.

Step 2: Check for Linearity

An equation is linear if it can be written in the form \( a_2(x) y'' + a_1(x) y' + a_0(x) y = f(x) \) where \( a_2(x), a_1(x), a_0(x) \) are functions of \( x \) only, and \( f(x) \) is a function independent of \( y, y', \) and \( y'' \). In the given equation, all coefficients of \( y'', y', \) and \( y \) are functions of \( x \), and \( 4y \) serves as a source term. The term \( 4y \) makes this equation nonlinear because it depends on the function \( y \).

Step 3: Verify the Presence of Linear Terms

Even though the arrangement \( y'' + (\sin x) y' - x y - 4y = 0 \) suggests linear coefficients, the presence of \( 4y \) on the right side means the equation itself includes \( y \) in non-linear interaction (as part of the inhomogeneous component), hence it disrupts the linearity.

Key Concepts

Linear Equations

Linear differential equations are a special category of equations where the unknown function and its derivatives are only multiplied by functions of the independent variable, not multiplying each other. This means an equation is linear if it can be expressed in the general form:

• \( a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \ldots + a_1(x) y' + a_0(x) y = f(x) \)

Here, \( a_n(x), a_{n-1}(x), \ldots, a_0(x) \) are coefficient functions dependent solely on the variable \( x \), and \( f(x) \) is a function that should not involve the unknown function \( y \), its derivatives, or powers of \( y \).
In our case, the equation is \( y'' + (\sin x) y' - x y = 4 y \). While it initially appears linear because the coefficients are functions of \( x \), the presence of \( 4y \) on the right-hand side indicates a nonlinear interaction. This is because \( y \) should not appear outside the differential terms for linearity as part of \( f(x) \).
Understanding whether an equation is linear helps in determining the methods used for finding solutions.

Nonlinear Equations

Nonlinear differential equations are characterized by having terms that are not just a sum of derivatives multiplied by functions of the independent variable. This can include products of the unknown function or its derivatives, and functions of these products. An equation exhibiting any such behavior is nonlinear.
For instance, the equation \( y'' + (\sin x) y' - x y = 4 y \) includes a non-linear interaction because \( 4y \) cannot be separated and put into a form that contains only the independent variable \( x \). This term makes the equation nonlinear by involving the function \( y \) in a manner that is not allowed in a linear equation.

• It complicates the equation and sometimes means we need different or more complex methods to find its solutions.
• Nonlinear equations often model real-world phenomena more accurately than linear equations.

Recognizing nonlinear forms is crucial since they often indicate a need for specialized solution techniques.

Homogeneous Equations

A key feature of linear differential equations is whether they are homogeneous or nonhomogeneous. A linear differential equation is homogeneous if the function \( f(x) = 0 \). This means that all terms are simply multiples of the unknown function and its derivatives, with no standalone forcing term or constant.
In contrast, an equation such as our example \( y'' + (\sin x) y' - x y = 4 y \) would be classified inhomogeneous if it were linear, because the \( 4y \) term does not drop to zero when rewritten in a standard homogeneous form.
Recognizing these equations, identifying if they have solutions without any constant terms or external inputs is essential:

• Homogeneous equations often focus solely on the internal dynamics of the system they model.
• Nonhomogeneous equations reflect systems influenced by external factors or inputs, shown through the non-zero \( f(x) \).

Understanding whether an equation is homogeneous affects the approach we take to solve it.