Problem 5 - A First Course in Differential Equations with Modeling Applications

Question

Find the general solution of the given second-order differential equation. $$y^{\prime \prime}+8 y^{\prime}+16 y=0$$

Step-by-Step Solution

Answer

The general solution is \( y(t) = c_1 e^{-4t} + c_2 t e^{-4t} \).

Step 1: Identify the type of differential equation

The given differential equation is a linear homogeneous second-order differential equation with constant coefficients: \( y'' + 8y' + 16y = 0 \).

Step 2: Write down the characteristic equation

The characteristic equation for a differential equation of the form \( ay'' + by' + cy = 0 \) is obtained by replacing \( y'' \) with \( m^2 \), \( y' \) with \( m \), and \( y \) with \( 1 \). Thus, the characteristic equation becomes \( m^2 + 8m + 16 = 0 \).

Step 3: Solve the characteristic equation

Factor the quadratic characteristic equation: \( m^2 + 8m + 16 = (m + 4)^2 = 0 \). This has a repeated root \( m = -4 \).

Step 4: Write the general solution

Since the characteristic equation has a repeated root, the general solution of the differential equation is given by: \( y(t) = c_1 e^{-4t} + c_2 t e^{-4t} \), where \( c_1 \) and \( c_2 \) are arbitrary constants.

Key Concepts

Linear Homogeneous Differential Equation

A linear homogeneous differential equation is a powerful tool in mathematics, mainly used to describe phenomena in physics and engineering. It is called 'linear' because every term involving the unknown function and its derivatives is linear, meaning that they are to the first power and multiplied by constants. The equation specified in this exercise, \( y'' + 8y' + 16y = 0 \), is homogeneous. This means every term depends on the function \( y \) and its derivatives, and the equation equals zero.

Key characteristics of these equations include:

• They can be of any order, but here, we deal with a second-order equation.
• No term is independent of the unknown function \( y \).
• The coefficients are constants.

Understanding the structure of these equations helps in simplifying and solving them systematically.

Characteristic Equation

The characteristic equation is crucial for solving linear homogeneous differential equations, especially those with constant coefficients. You derive it by substituting derivatives with a radical term \( m \). For example, in our given equation \( y'' + 8y' + 16y = 0 \), substitute \( y'' \) with \( m^2 \), \( y' \) with \( m \), and \( y \) with 1, resulting in the characteristic equation \( m^2 + 8m + 16 = 0 \).

Here's a quick breakdown of this process:

• Each \( m \) represents an order of derivative, so \( m^2 \) mimics the second derivative \( y'' \).
• The form of the characteristic equation is targeted at simplifying higher-order differential equations into an easily solvable algebraic equation.
• Its roots tell us an essential part of the solution's form.

Solving the characteristic equation gives information about the behavior and appearance of the function \( y \). Keep in mind that the nature of roots (real or complex, distinct or repeated) influences the solution structure.

General Solution of Differential Equations

Once you have the roots of the characteristic equation, you can construct the general solution of the differential equation. In this example, solving \( m^2 + 8m + 16 = 0 \) gives a repeated root, \( m = -4 \). This means that the solutions are not just simple exponentials but require a modification to accommodate the repetition.

The formation from roots to solutions follows these rules:

• If roots are real and distinct, each root contributes a solution of the form \( c_i e^{m_i t} \), where \( c_i \) is a constant.
• If roots are repeated, as in this case, they contribute solutions like \( c_i e^{m_i t} + c'_i t e^{m_i t} \).
• The constants \( c_1 \), \( c_2 \), etc., represent arbitrary constants determined based on initial conditions or other constraints.

Thus, the general solution here is \( y(t) = c_1 e^{-4t} + c_2 t e^{-4t} \). This reflects both the presence of repeating roots and the essential features of the underlying differential equation. The constants \( c_1 \) and \( c_2 \) will be determined when additional conditions are provided.