Problem 15 - A First Course in Differential Equations with Modeling Applications

Question

Find a linear differential operator that annihilates the given function. $$1+6 x-2 x^{3}$$

Step-by-Step Solution

Answer

The operator that annihilates the function is \(\frac{d^3}{dx^3}\).

Step 1: Identify the Function Type

The function given is a polynomial: \(P(x) = 1 + 6x - 2x^3\). It is a cubic polynomial because the highest power of \(x\) is 3.

Step 2: General Form of Differential Operator

For a polynomial of degree n, the differential operator that annihilates it is the nth derivative. In this case, since \(n = 3\), we use the third derivative: \(\frac{d^3}{dx^3}\).

Step 3: Apply Differential Operator

Apply the third derivative to the polynomial. Calculate \(\frac{d^3}{dx^3}(1 + 6x - 2x^3)\).

Step 4: First Derivative Calculation

Find the first derivative: \(P'(x) = \frac{d}{dx}(1 + 6x - 2x^3) = 0 + 6 - 6x^2\).

Step 5: Second Derivative Calculation

Find the second derivative: \(P''(x) = \frac{d^2}{dx^2}(6 - 6x^2) = -12x\).

Step 6: Third Derivative Calculation

Find the third derivative: \(P'''(x) = \frac{d^3}{dx^3}(-12x) = -12\).

Step 7: Verify Annihilation

A constant function is successfully annihilated by the third derivative operator. Therefore, the operator \(\frac{d^3}{dx^3}\) annihilates the function \(1+6x-2x^3\).

Key Concepts

Polynomial Functions

Polynomial functions are a class of functions that resemble multi-variable algebraic expressions. They consist of variables, coefficients, and exponents that are whole numbers. These expressions are combined through addition, subtraction, and multiplication. Commonly seen as expressions like \(ax^n + bx^{n-1} + ... + k\), polynomial functions can be different degrees:

• Constant Polynomial: No variable terms, only a constant, like \(7\).
• Linear Polynomial: Exponents of 1, such as \(3x + 2\).
• Quadratic Polynomial: Exponents up to 2, like \(5x^2 + 3x + 1\).
• Cubic Polynomial: Exponents up to 3, for example, \(2x^3 - x^2 + 3\).

For the given exercise, the function \(1 + 6x - 2x^3\) is a cubic polynomial. This tells us that the highest power of \(x\) is 3, which is essential to know as it guides how we choose the correct differential operator to annihilate it.

Differential Operators

Differential operators are tools used in calculus to perform differentiation on functions. They help us investigate how functions change by taking derivatives. These operators are denoted by symbols like \(\frac{d}{dx}\) and can act on a variety of functions.

• First Derivative: Indicates how the function changes at each point. A differential operator \(\frac{d}{dx}\) is used.
• Higher Derivatives: Produced by applying the differential operator multiple times, like \(\frac{d^2}{dx^2}\) for the second derivative, etc.

In our context, the function is cubic, thus requiring the third derivative to annihilate it. This uses the operator \(\frac{d^3}{dx^3}\). By applying it to the polynomial \(1 + 6x - 2x^3\), we ensure that all terms become zero, concluding the annihilation process.

Derivatives

Derivatives are a fundamental concept in calculus and measure how a function changes as its input changes. Calculating derivatives involves differentiating a function to determine its rate of change.
The step-by-step solution provided employs the following stages:

• Finding the First Derivative: \(P'(x) = 6 - 6x^2\). This involves using basic differentiation rules.
• Finding the Second Derivative: \(P''(x) = -12x\). This omits constants because their derivatives are zero.
• Finding the Third Derivative: \(P'''(x) = -12\). As a result, this becomes a constant and effectively concludes the annihilation process.

By the time we reach the third derivative, the presence of a constant confirms that the operator \(\frac{d^3}{dx^3}\) successfully annihilates the function. Once a constant is reached, further derivatives result in zero, hence proving annihilation is complete.