Problem 33 - A First Course in Differential Equations with Modeling Applications

Question

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$\begin{aligned}&y^{\prime \prime}-4 y^{\prime}+4 y=2 e^{2 x}+4 x-12\\\&y=c_{1} e^{2 x}+c_{2} x e^{2 x}+x^{2} e^{2 x}+x-2,(-\infty, \infty) \end{aligned}$$

Step-by-Step Solution

Answer

The function satisfies the differential equation, confirming it is the general solution.

Step 1: Verify the General Form of the Solution

The problem asks us to verify whether the given function \( y = c_{1} e^{2x} + c_{2} x e^{2x} + x^{2} e^{2x} + x - 2 \) is a solution to the differential equation \( y'' - 4y' + 4y = 2 e^{2x} + 4x - 12 \). The solution consists of a complementary solution to the associated homogeneous equation plus a particular solution to the nonhomogeneous equation.

Step 2: Differentiate the Function

Compute the first derivative of the function \( y \). Use the power rule and product rule for differentiating each term:\[ y' = c_{1} (2e^{2x}) + c_{2} (2xe^{2x} + e^{2x}) + (2x^2 e^{2x} + 2 x e^{2x}) + 1 \]Simplifying gives:\[ y' = 2c_{1} e^{2x} + c_{2} (2xe^{2x} + e^{2x}) + (2xe^{2x} (x + 1) + 2 x e^{2x}) + 1 \].

Step 3: Differentiate Again for the Second Derivative

Compute the second derivative \( y'' \):\[ y'' = c_{1} (4e^{2x}) + c_{2} (4xe^{2x} + 4e^{2x}) + (4x^2 e^{2x} + 8xe^{2x} + 4e^{2x}) \].

Step 4: Substitute into the Differential Equation

Insert expressions for \( y, y', \) and \( y'' \) into the differential equation:\[ y'' - 4y' + 4y = (c_{1} e^{2x} + c_{2} x e^{2x} + x^2 e^{2x} + x - 2)'' - 4(c_{1} e^{2x} + c_{2} x e^{2x} + x^2 e^{2x} + x - 2)' + 4(c_{1} e^{2x} + c_{2} x e^{2x} + x^2 e^{2x} + x - 2) \].

Step 5: Simplify and Solve

Simplify the left-hand side of the equation to match the given nonhomogeneous part of the equation (\(2 e^{2x} + 4x - 12\)). After simplifying both sides, each corresponding term on both sides of the equation should be equal, verifying that \( y = c_{1} e^{2x} + c_{2} x e^{2x} + x^2 e^{2x} + x - 2 \) is indeed a solution.

Key Concepts

Nonhomogeneous Differential Equation

Differential equations that include terms with no derivatives are categorized as nonhomogeneous differential equations. These equations typically take the form:\[ ay'' + by' + cy = g(x) \]where the function \( g(x) \) is known as the nonhomogeneous part or external force. This means the equation isn't strictly derivative-based and involves an additional function, affecting the system described.
Nonhomogeneous differential equations are prevalent when modeling real-world dynamics such as forced oscillations in mechanical systems or electrical circuits under external voltage sources. The presence of \( g(x) \) differentiates them from homogeneous differential equations, which have \( g(x) = 0 \).
Solving these equations involves a combination of complementary and particular solutions. This separation allows addressing each part of the equation efficiently and clearly. Understanding nonhomogeneous differential equations expands your ability to solve complex, real-world problems elegantly.

General Solution

The general solution of a nonhomogeneous differential equation is a crucial concept encompassing all possible solutions. This solution, expressed as:\[ y = y_c + y_p \]where \( y_c \) is the complementary function and \( y_p \) is the particular solution, offers a complete framework to resolve the differential equation.

• Complementary Function (CF): Solves the associated homogeneous equation \( ay'' + by' + cy = 0 \).
• Particular Solution (PS): Solves the entire nonhomogeneous equation, including \( g(x) \).

By combining these components, the general solution not only satisfies the specific differential equation but also accommodates boundary conditions for unique solutions. Knowing the distinction and combination of CF and PS is fundamental in unraveling various instances of differential equations. This process exemplifies the systematic approach needed in complex mathematics, emphasizing integration of theory and method.

Complementary Function

The complementary function (CF) is derived from solving the associated homogeneous equation \( ay'' + by' + cy = 0 \). This component of the general solution represents the intrinsic behavior of the equation without the influence of the external force \( g(x) \).
To find the CF, you typically:

• Assume a solution of the form \( y = e^{rx} \).
• Substitute this into the homogeneous equation to find the characteristic equation.
• Solve for the roots \( r \) of the characteristic equation, which are often real or complex numbers.

These roots determine the form of the complementary function:

• Real and distinct roots lead to solutions like \( c_1 e^{r_1 x} + c_2 e^{r_2 x} \).
• Repeated roots require multiplying solutions by increasing powers of \( x \), such as \( c_1 e^{rx} + c_2 x e^{rx} \).
• Complex roots bring sine and cosine into play, forming solutions like \( e^{ ext{Re}(r)x} (c_1 \cos( ext{Im}(r)x) + c_2 \sin( ext{Im}(r)x)) \).

The CF essentially captures the natural response of the system and is crucial for predicting behaviors under no external forces.

Particular Solution

The particular solution (PS) focuses on satisfying the nonhomogeneous aspect of the differential equation. Unlike the complementary function, which handles the homogeneous part, the particular solution directly accounts for the influence of the non-zero function \( g(x) \).
Finding the PS involves:

• Choosing an appropriate method, such as the method of undetermined coefficients or variation of parameters, to guess the form of \( y_p \).
• Substituting this guessed form into the nonhomogeneous differential equation.
• Solving for any unknown coefficients to fit the entire equation accurately.

For example, if \( g(x) \) is a polynomial, exponential, sine, or cosine, similar functions can be used in the particular solution, adjusting for any elements already present in the CF.
Completing the particular solution effectively integrates the external influences into the differential equation, allowing it to accurately model real-world systems with forcing functions. This step is vital as it permits accurate prediction and manipulation of systems subjected to external conditions.