Problem 26 - A First Course in Differential Equations with Modeling Applications

Question

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$4 y^{\prime \prime}-4 y^{\prime}+y=0 ; \quad e^{x / 2}, x e^{x / 2},(-\infty, \infty)$$

Step-by-Step Solution

Answer

The general solution is \( y(x) = C_1 e^{x/2} + C_2 x e^{x/2} \) on \((-\infty, \infty)\).

Step 1: Verify Linear Independence

To verify that the functions \( e^{x / 2} \) and \( x e^{x / 2} \) are linearly independent, we need to compute the Wronskian determinant. Let \( y_1 = e^{x / 2} \) and \( y_2 = x e^{x / 2} \). We calculate their derivatives: \( y_1' = \frac{1}{2} e^{x / 2} \) and \( y_2' = e^{x / 2} + \frac{1}{2}x e^{x / 2} \). The Wronskian \( W(y_1, y_2) \) is given by:\[W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \ y_1' & y_2' \end{vmatrix} = \begin{vmatrix} e^{x/2} & x e^{x/2} \ \frac{1}{2} e^{x/2} & \left(1 + \frac{x}{2}\right) e^{x/2} \end{vmatrix}\]Simplifying, we find:\[W(y_1, y_2) = e^{x} \left(1 + \frac{x}{2}\right) - \frac{x}{2} e^{x} = e^{x} eq 0\]The Wronskian is non-zero, so the functions are linearly independent on the interval \((-\infty, \infty)\).

Step 2: Confirm Solutions to the Differential Equation

Substitute each function into the differential equation \(4 y'' - 4 y' + y = 0\) to ensure they are solutions.* For \( y = e^{x / 2} \): * \( y' = \frac{1}{2} e^{x / 2}, y'' = \frac{1}{4} e^{x / 2} \) * Substitute: \( 4 \cdot \frac{1}{4} e^{x / 2} - 4 \cdot \frac{1}{2} e^{x / 2} + e^{x / 2} = 0 \), which simplifies to \( e^{x / 2} - 2 e^{x / 2} + e^{x / 2} = 0 \).* For \( y = x e^{x / 2} \): * \( y' = e^{x / 2} + \frac{1}{2} x e^{x / 2}, y'' = \frac{1}{2} e^{x / 2} + \frac{1}{2} e^{x / 2} + \frac{1}{4} x e^{x / 2} \) * Substitute: \( 4 (\frac{1}{2} e^{x / 2} + \frac{1}{2} e^{x / 2} + \frac{1}{4} x e^{x / 2}) - 4 (e^{x / 2} + \frac{1}{2} x e^{x / 2}) + x e^{x / 2} = 0 \), simplifies to \(x e^{x / 2} = x e^{x / 2} \).Both functions satisfy the differential equation.

Step 3: Form General Solution

Given two linearly independent solutions \( y_1 = e^{x / 2} \) and \( y_2 = x e^{x / 2} \) to a second-order linear homogeneous differential equation, the general solution is a linear combination of these solutions:\[y(x) = C_1 e^{x / 2} + C_2 x e^{x / 2}\]where \( C_1 \) and \( C_2 \) are arbitrary constants.

Key Concepts

Linear Independence

Linear independence is a fundamental concept when dealing with sets of functions and differential equations. In the context of differential equations, we are often interested in finding out if a set of functions can serve as a basis for the solution space of the equation. If two or more functions are linearly independent, it essentially means none of the functions in the set can be written as a linear combination of the others. This characteristic is crucial for forming a complete solution set for differential equations.

To check for linear independence, you typically use the Wronskian determinant, which involves calculating derivatives. If the Wronskian of a set of functions is non-zero at some point in the interval, the functions are considered linearly independent over that interval. These functions can then be used to express the general solution of a corresponding homogeneous differential equation.

Wronskian

The Wronskian is a determinant used for determining the linear independence of a set of functions. For two functions, say \( y_1 \) and \( y_2 \), the Wronskian \( W(y_1, y_2) \) is computed as follows:

• Write a matrix with the first row as \( y_1 \) and \( y_2 \), and the second row as their respective derivatives \( y_1' \) and \( y_2' \).
• Calculate the determinant of this 2x2 matrix.

The resulting determinant tells us about the nature of linear independence. If the Wronskian is never zero on the interval of interest, the functions are linearly independent.For the functions \( e^{x/2} \) and \( x e^{x/2} \) in the exercise, the Wronskian calculation showed a non-zero result, indicating these functions are linearly independent on \((-\infty, \infty)\). This step is crucial since it confirms that both functions can be used to form the general solution of the differential equation.

General Solution

Finding the general solution to a differential equation is like discovering a "formula" that encompasses all possible specific solutions to the problem. For a second-order linear homogeneous differential equation, the general solution is a linear combination of its linearly independent solutions.

In our exercise, once it is established that \( e^{x/2} \) and \( x e^{x/2} \) are linearly independent solutions, we form the general solution by combining them with arbitrary constants \( C_1 \) and \( C_2 \):\[y(x) = C_1 e^{x/2} + C_2 x e^{x/2}\]This expression accounts for all initial conditions and particular behaviors the solution might exhibit across the interval \((-\infty, \infty)\). The constants \( C_1 \) and \( C_2 \) can be adjusted based on additional conditions or constraints given in a problem, allowing us to fit the solution to specific scenarios.

Homogeneous Differential Equation

A homogeneous differential equation is a specific type of equation where every term is a function of the dependent variable and its derivatives. For these equations, every term can directly trace back to the function itself, with no external or added functions.

In simpler terms, a homogeneous differential equation is one that is consistent and self-contained, without any standalone terms—unlike non-homogeneous equations which include external inputs or non-zero constants.In the provided example, the differential equation \(4 y'' - 4 y' + y = 0\) is homogeneous because all its components are derivatives of \( y \) or the function \( y \) itself. Solving such equations often involves finding solutions that are multiples of the given functions, which are then used to construct the general solution, as we have seen with the linearly independent solutions \( e^{x/2} \) and \( x e^{x/2} \). This approach is fundamental in solving many physical and mathematical problems where the governing equations are inherently homogeneous.