Problem 9 - Advanced Engineering Mathematics

Question

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-y^{\prime}=-3\)

Step-by-Step Solution

Answer

The general solution is \( y = C_1 + C_2 e^x - 3 \).

Step 1: Identify the Homogeneous Equation

Separate the homogeneous part of the given differential equation, ignoring the constant term. The homogeneous equation is given by:\[ y'' - y' = 0 \]

Step 2: Solve the Homogeneous Equation

Solve the characteristic equation derived from the homogeneous part:\[ m^2 - m = 0 \]To solve, factor the equation:\[ m(m-1) = 0 \]The solutions are \( m = 0 \) and \( m = 1 \). Thus, the general solution to the homogeneous equation is:\[ y_h = C_1 + C_2 e^x \]

Step 3: Identify the Particular Solution Form

The non-homogeneous term on the right is a constant, \(-3\). We'll assume a particular solution, \( y_p \), in the form of a constant:\[ y_p = A \]

Step 4: Determine the Particular Solution

Substitute \( y_p = A \) into the original differential equation.First, find the derivatives:\[ y_p' = 0, \quad y_p'' = 0 \]Substitute these into the equation:\[ 0 - 0 = -3 \]Since the left side must equal \(-3\), we find that \( A = -3 \).Thus, the particular solution is:\[ y_p = -3 \]

Step 5: Form the General Solution

The general solution to the differential equation is the sum of the homogeneous solution and the particular solution:\[ y = y_h + y_p = C_1 + C_2 e^x - 3 \]

Key Concepts

Differential Equations

Differential equations provide a way to describe how quantities change with respect to one another. They are equations that relate a function to its derivatives. In our exercise, the differential equation is given by:\[ y'' - y' = -3 \]Here, \( y'' \) is the second derivative and \( y' \) is the first derivative of some unknown function \( y \). The left side of the equation describes the behavior of these derivatives, while the right side includes a constant term, \(-3\), which represents an external condition or force acting on the system.

• Key components: Derivatives (\( y' \), \( y'' \)) indicate how the function \( y \) changes at different levels.
• Constant term: External influence, unaffected by the derivative conditions.

Understanding the structure of a differential equation helps us distinguish between different types of solutions: the homogeneous and the particular solutions.

Particular Solution

In the world of differential equations, a particular solution is a specific solution that satisfies the equation for given conditions outside the zero context of the homogeneous equation. It's essentially how the equation adjusts to external forces, represented by the non-zero terms on the equation's right side.
In our exercise, we have a constant term \(-3\), so we assume a particular solution\( y_p \) of this form:\[ y_p = A \]When you substitute this constant back into the original equation,\[ 0 - 0 = -3 \]we find that \( A \) must be \(-3\). Thus, the particular solution is simply:\[ y_p = -3 \]

• Purpose: Adjusts the differential equation to specific external conditions.
• Method: Often involves guessing a form (using undetermined coefficients) that matches the structure of external terms.

Understanding particular solutions is essential for capturing the effects of additional conditions imposed on a system.

Homogeneous Equation

A homogeneous equation is a differential equation set to zero, stripping away external influences represented by non-zero terms. This simplifies the equation, focusing on the intrinsic properties of the function itself. From our example, the homogeneous equation derived is:\[ y'' - y' = 0 \]Solving this requires finding solutions to the characteristic equation derived from the homogeneous part, which includes roots that determine the function's behavior:\[ m^2 - m = 0 \]Factoring gives:\[ m(m-1) = 0 \]This yields roots \( m = 0 \) and \( m = 1 \), leading to a general solution for the homogeneous equation:\[ y_h = C_1 + C_2 e^x \]

• Key characteristic: The solution involves arbitrary constants (\( C_1, C_2 \)), highlighting internal behavior without external force.
• Importance: Forms the basis for the total solution when combined with the particular solution.

Understanding the homogeneous equation is crucial, as it lays the groundwork before external conditions are applied to derive a full solution.