Problem 18 - Differential Equations and Linear Algebra

Question

Write the given nonhomogeneous differential equation as an operator equation, and give the associated homogeneous differential equation. $$y^{\prime \prime}+4 x y^{\prime}-6 x^{2} y=x^{2} \sin x$$

Step-by-Step Solution

Answer

The given nonhomogeneous differential equation is rewritten as an operator equation: \((D^2 + 4xD - 6x^2)y = x^2 \sin x\). The associated homogeneous differential equation is \((D^2 + 4xD - 6x^2)y = 0\).

Step 1: Identify the nonhomogeneous differential equation

The given nonhomogeneous differential equation is: \[y^{\prime \prime}+4 x y^{\prime}-6 x^{2} y=x^{2} \sin x\] Step 2: Rewrite the equation as an operator equation

Step 2: Rewrite as an operator equation

The given differential equation can be rewritten using the differential operator, D, where \(D^n = \frac{d^n}{dx^n}\), and \(n\) is the order of the derivative. For this equation, we have: \[(D^2 + 4xD - 6x^2)y = x^2 \sin x\] Step 3: Write down the associated homogeneous differential equation

Step 3: Find the homogeneous differential equation

The associated homogeneous differential equation is obtained by setting the nonhomogeneous term to zero. In this case, we have: \[(D^2 + 4xD - 6x^2)y = 0\] So, the given nonhomogeneous differential equation as an operator equation is \((D^2 + 4xD - 6x^2)y = x^2 \sin x\), and the associated homogeneous differential equation is \((D^2 + 4xD - 6x^2)y = 0\).

Key Concepts

Differential Operator

A differential operator is a tool used in calculus to simplify and solve differential equations. It's like a shorthand notation that helps express operations involving derivatives. In the context of the given differential equation, a differential operator, denoted by \( D \), takes the form \( D^n = \frac{d^n}{dx^n} \). This means that \( D^2 \) represents the second derivative, \( D^1 \) the first derivative, and so on. By using a differential operator, you can rewrite a complex-looking differential equation in a more compact form. This makes the process of solving differential equations much more manageable. For instance, the differential expression \( y'' + 4xy' - 6x^2y \) can be expressed as the operator equation \( (D^2 + 4xD - 6x^2)y \). This shift to operators helps simplify solving the equation, especially when dealing with higher-order derivatives.

Homogeneous Differential Equation

Homogeneous differential equations are those where all terms can be attributed directly to the variable and its derivatives, without any additional terms or functions. In simpler words, there is no external function, or source, affecting the equation.In the context of the exercise, transforming a nonhomogeneous differential equation into its homogeneous counterpart involves removing the nonhomogeneous part. This part is typically found on the right side of the equation and consists of terms not directly related to the dependent variable or its derivatives. For example, dropping the \( x^2 \sin x \) in the operator equation \( (D^2 + 4xD - 6x^2)y = x^2 \sin x \) results in the homogeneous version \( (D^2 + 4xD - 6x^2)y = 0 \). Analyzing homogeneous equations is critical because they represent the natural behavior of the system without external forces.

Operator Equation

An operator equation utilizes operators, like the differential operator, to express and solve equations involving derivatives. In the study of differential equations, converting the equations to operator form is a common approach, as it streamlines both the expression and solving processes.With operator equations, the equation's structure relates 'operations' on the function rather than simply stating 'actions' or 'modifications'. This type of equation also makes it easier to apply various methods and techniques to find solutions.Take the nonhomogeneous equation \( y'' + 4xy' - 6x^2y = x^2 \sin x \), which is rewritten as an operator equation: \( (D^2 + 4xD - 6x^2)y = x^2 \sin x \). The operator equation format highlights the operations performed on \( y \), making it more apparent which methods are suitable for solution.