Problem 9 - Advanced Engineering Mathematics

Question

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-y^{\prime}=-3 $$

Step-by-Step Solution

Answer

The general solution to the differential equation is: \( y = C_1 + C_2e^x + 3x \).

Step 1: Identify the Type of Differential Equation

The given equation is a second-order linear differential equation with constant coefficients: \( y'' - y' = -3 \). This is a non-homogeneous equation.

Step 2: Solve the Homogeneous Equation

First, find the complementary solution by solving the homogeneous equation: \( y'' - y' = 0 \). The characteristic equation is: \( r^2 - r = 0 \). Factoring gives: \( r(r - 1) = 0 \). So, the roots are \( r = 0 \) and \( r = 1 \). The complementary solution is: \( y_c = C_1 + C_2e^x \), where \( C_1 \) and \( C_2 \) are constants.

Step 3: Determine Particular Solution Form

Since the non-homogeneous term is a constant, \( -3 \), a particular solution \( y_p \) can be guessed as a constant: \( y_p = A \). Therefore, substitute \( y_p \) into the differential equation to find \( A \).

Step 4: Substitute and Solve for the Particular Solution

Substituting \( y_p = A \) into \( y'' - y' = -3 \) results in: \( 0 - 0 = -3 \),which implies that our initial guess was incorrect. Instead, try replacing \( A \) with a linear function, \( y_p = Ax \). This results in: \( 0 - A = -3 \), thus, \( A = 3 \). So, the particular solution is \( y_p = 3x \).

Step 5: Write the General Solution

Combine the complementary solution and the particular solution: The general solution is: \( y = y_c + y_p = C_1 + C_2e^x + 3x \).

Key Concepts

Undetermined Coefficients

The method of undetermined coefficients is a handy tool used to solve non-homogeneous linear differential equations. It works especially well when the non-homogeneous part, the right side of the equation, is of a simple form like a polynomial, exponential, or sine and cosine function. Here's a brief explanation of how it works:

• First, we solve the homogeneous equation to find the complementary solution, which accounts for the natural behavior of the system described by the differential equation.
• Next, we make an educated guess for the particular solution based on the form of the non-homogeneous term. This initial guess often involves undetermined constants, hence the name "undetermined coefficients."
• Afterwards, we substitute our guess into the original equation and solve for these constants to satisfy the equation exactly.

Because this method relies on making an assumption about the form of the solution, it's only applicable when the function type is predictable, like the ones mentioned. In our example, we initially guessed a constant solution. When this didn’t work, we tried a linear function instead, which eventually gave us the correct particular solution.

Second-Order Linear Differential Equations

A second-order linear differential equation is identified by the presence of the second derivative in its formulation. It follows a standard form such as:\[ a y'' + b y' + c y = f(x) \]where \(a\), \(b\), and \(c\) are constants, and \(f(x)\) is a function of that is often a non-zero term, making the equation non-homogeneous.Key characteristics of these equations include:

• The degree of the derivative: The highest derivative present is the second derivative, \(y''\).
• Linearity: All terms containing the function \(y\) and its derivatives are linear.
• Constant coefficients: (when applying undetermined coefficients) coefficients \(a\), \(b\), and \(c\) do not depend on \(x\).

For example, in our problem, the differential equation is given by \( y'' - y' = -3 \), making it a second-order equation. Solving such equations typically involves finding the complementary function by addressing the associated homogeneous equation where \(f(x) = 0\), followed by solving for the particular solution. By combining both solutions, you get the overall solution showcasing all possible behaviors of the system around the conditions defined.

Non-Homogeneous Differential Equations

Non-homogeneous differential equations are those that include a term that isn't a function of the dependent variable or its derivatives alone. Simply put, there is typically an added external component or forcing term, indicated as \(f(x)\) in the general equation \(a y'' + b y' + c y = f(x)\).These equations are vital because they model many real-world scenarios where systems are subject to external influences. Understanding the process of solving them includes two primary parts:

• Finding the complementary solution: Solve the associated homogeneous equation (where \(f(x)\) is set to 0) as described with undetermined coefficients.
• Finding the particular solution: This involves identifying a specific solution to the non-homogeneous equation that accounts for the external factor.

In the context of our original problem, this meant managing both the homogeneous part \(y'' - y' = 0\) as well as the constant term \(-3\), which was external to the system's inherent properties.The general solution is constructed as a sum of the complementary function and the particular solution expressing the overall system response, including responses to external forces.