Problem 30 - Advanced Engineering Mathematics

Question

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$ y^{(4)}+y^{\prime \prime}=0 ; 1, x, \cos x, \sin x,(-\infty, \infty) $$

Step-by-Step Solution

Answer

The functions form a fundamental set; the general solution is \( y = C_1 + C_2 x + C_3 \cos x + C_4 \sin x \).

Step 1: Understanding the Differential Equation

The given differential equation is a fourth-order linear homogeneous differential equation: \( y^{(4)} + y'' = 0 \). We need to verify if the functions \( 1, x, \cos x, \sin x \) form a fundamental set of solutions for this equation over the interval \( (-\infty, \infty) \).

Step 2: Checking Linear Independence

To verify if these functions form a fundamental set of solutions, they must be linearly independent. Calculate the Wronskian determinant of the functions: \( W(1, x, \cos x, \sin x) \). If the Wronskian is non-zero for some \( x \) in the interval, the functions are linearly independent.

Step 3: Computation of the Wronskian

Compute the Wronskian: \[W(1, x, \cos x, \sin x) = \begin{vmatrix}1 & x & \cos x & \sin x \0 & 1 & -\sin x & \cos x \0 & 0 & -\cos x & -\sin x \0 & 0 & \sin x & -\cos x\end{vmatrix}.\]Simplifying the determinant, we find it is non-zero, confirming linear independence.

Step 4: Verify the Solutions for the DE

Verify that each function satisfies the differential equation. 1. The fourth derivative of \(1\) and \(x\) is zero, and their second derivatives are zero, so they satisfy the equation.2. The fourth derivative of \(\cos x\) is \(\cos x\), and its second derivative is \(-\cos x\); hence, \(\cos x\) satisfies the equation.3. Similarly, the fourth derivative of \(\sin x\) is \(\sin x\), and its second derivative is \(-\sin x\); thus, \(\sin x\) satisfies the equation.

Step 5: Forming the General Solution

With the confirmed fundamental set \( \{ 1, x, \cos x, \sin x \} \), the general solution for the differential equation is a linear combination of these functions: \[ y(x) = C_1 + C_2 x + C_3 \cos x + C_4 \sin x, \]where \( C_1, C_2, C_3, C_4 \) are arbitrary constants.

Key Concepts

Fundamental Set of Solutions

When solving higher-order differential equations, one of the key steps is identifying a fundamental set of solutions. A fundamental set effectively gives us a basis for the solution space of the differential equation. In simpler terms, it is a group of solutions that can be combined to form the most general solution for the equation. For a set of solutions to be deemed 'fundamental,' they must be linearly independent. This ensures they provide a complete representation of the solution space.

In the context of the exercise, the functions \(1, x, \cos x, \sin x\) were considered to form a fundamental set of solutions for the given differential equation. They together cover various behaviors of the potential solutions, such as constant, linear, and trigonometric forms.

Wronskian Determinant

The Wronskian determinant is an essential tool in determining whether a group of functions is linearly independent. To calculate it, we construct a matrix where each row is composed of the derivatives of the functions being tested. The determinant of this matrix is the Wronskian. If a Wronskian is non-zero at any point in the interval, the functions are linearly independent.

For the functions \(1, x, \cos x, \sin x\), calculating the Wronskian yields a non-zero value, indicating these functions are linearly independent. Computing the determinant involves differentiating each function up to the third derivative and organizing these into a matrix. Simplifying the determinant shows it is not zero, confirming our functions are a fundamental set.

Linear Independence

Linear independence between functions means that no function in the set can be written as a linear combination of the others. This is crucial when forming a fundamental set of solutions because it confirms the uniqueness and completeness of the set.

By using the Wronskian determinant, it is straightforward to test for linear independence. In this exercise, the fact that the determinant of the matrix \(W(1, x, \cos x, \sin x)\) is non-zero confirms that these functions cannot be written in terms of each other. Therefore, they span the solution space of the differential equation without any redundancies.

General Solution of Differential Equations

The general solution of a differential equation is a combination of all possible solutions expressed in their simplest form. It represents the entire solution space and includes arbitrary constants, which can be adjusted based on initial conditions or specific requirements.

For the given fourth-order differential equation \(y^{(4)} + y'' = 0\), with the verified fundamental set \( \{1, x, \cos x, \sin x\} \), the general solution can be constructed by taking a linear combination of these functions. Thus, the solution is given by:

\[ y(x) = C_1 + C_2 x + C_3 \cos x + C_4 \sin x \]

Here, \(C_1, C_2, C_3,\) and \(C_4\) are arbitrary constants that allow the general solution to meet specific boundary or initial conditions.