Problem 33 - Advanced Engineering Mathematics

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 given function is the verified general solution for the differential equation.

Step 1: Understand the Differential Equation and Interval

The given nonhomogeneous differential equation is \( y'' - 4y' + 4y = 2e^{2x} + 4x - 12 \). This needs to be solved for the interval \((-fty, fty)\). We are asked to verify a general solution of the form \( y = c_1 e^{2x} + c_2 xe^{2x} + x^2 e^{2x} + x - 2 \).

Step 2: Identify Particular Solution for Nonhomogeneous Equation

Given the nonhomogeneous part \(2e^{2x} + 4x - 12\), we examine the form of the particular solution. The solution for \(2e^{2x}\) is proportional to \(Ae^{2x}\), for \(4x\) it is proportional to \(Bx + C\), and for \(-12\) simply a constant \(D\).

Step 3: Test General Solution in Differential Equation

Substitute the general solution \(y = c_1 e^{2x} + c_2 xe^{2x} + x^2 e^{2x} + x - 2\) into the differential equation. Calculate \(y'\), \(y''\), and substitute them into the equation to check if it simplifies to \(2e^{2x} + 4x - 12\).

Step 4: Differentiate the General Solution

Calculate the first derivative \(y' = 2c_1 e^{2x} + c_2 e^{2x} + 2c_2 xe^{2x} + 2xe^{2x} + 2x^2 e^{2x} + 1\). Now compute the second derivative \(y'' = 4c_1 e^{2x} + 2c_2 e^{2x} + 4c_2 xe^{2x} + 4xe^{2x} + 2x^2 e^{2x}\).

Step 5: Substitute and Simplify

Substitute \(y\), \(y'\), and \(y''\) into the differential equation \(y'' - 4y' + 4y\). Simplify the terms to check if they produce the nonhomogeneous part \(2e^{2x} + 4x - 12\). Verify that constants \(c_1\), \(c_2\), and additional terms balance to account for solutions based on different parts of the nonhomogeneous term.

Step 6: Conclusion on General Solution

After substitution and simplification, confirm that the left-hand side of the differential equation equals the right-hand side \(2e^{2x} + 4x - 12\). Therefore, the given solution is verified as the general solution for the differential equation over the interval \((-fty, fty)\).

Key Concepts

General Solution

A general solution of a differential equation is a solution that contains all possible solutions of the equation. It typically involves arbitrary constants, which allow it to cover particular solutions. In the case of the given differential equation, \( y'' - 4y' + 4y = 2e^{2x} + 4x - 12 \), the general solution proposed is \(y = c_1 e^{2x} + c_2 xe^{2x} + x^2 e^{2x} + x - 2\). Notice here how

• \(c_1 e^{2x} + c_2 xe^{2x}\) represents the homogeneous solution part, solving \(y'' - 4y' + 4y = 0\)
• \(x^2 e^{2x} + x - 2\) is the particular specific adjustment to solve the nonhomogeneous equation.

This structure neatly adds a particular adaptation to the general case, ultimately encompassing both basic and unique, specific solutions.

Particular Solution

A particular solution addresses and solves the specific nonhomogeneous part of a differential equation that arises due to a non-zero value on the right-hand side. For our original problem, the nonhomogeneous part is expressed as \(2e^{2x} + 4x - 12\).

• The term \(2e^{2x}\) suggests a particular solution that involves a simple exponential adjustment, in this case, proportional to \(Ae^{2x}\).
• The linear term \(4x\) along with the constant \(-12\) suggests solutions that are linear and constant, represented as \(Bx + C\).

Together, these create a specific function that, when inserted into the device of the equation, reproduces precisely the surprising non-zero side. This fitting choice adds to a composite function, compatible with the rest of the general solution, assuring both consistency and correctness.

Differential Equation Verification

Verification of solutions is integral to ensure that they truly satisfy a given differential equation. To prove that the proposed general solution is accurate, it involves computing derivatives and substititating back into the equation. Here's how it unfolds for our differential equation. First, derive and compute both the first and second derivatives from the provided general solution \

• First derivative \(y'\), capturing change rates, involves computing derivatives of each component separately.
• Similarly, the second derivative \(y''\) provides the rate of change of the rate of change.

After calculating these, substituting the functions \(y\), \(y'\), and \(y''\) back into the differential equation is essential. Upon doing this, the result should match the non-zero right hand side, \(2e^{2x} + 4x - 12\), indicating the solution is accurate. It's a systematic check that involves algebraic manipulation and confirmation.

Two-Parameter Family of Functions

A two-parameter family of functions is a group of solutions that can be adjusted by varying two separate constants. In the context of our differential equation, these constants are \(c_1\) and \(c_2\), which allow any valid solution within this family. Each choice of these parameters provides us with a distinct, particular solution.

• \(c_1\) corresponds to scaling adjustments of the term \(e^{2x}\).
• \(c_2\) allows a similar modulation of the term \(xe^{2x}\).

This adjustable format means that from this family, an infinite multitude of solutions can be derived. Such flexibility is powerful, enabling an intricate yet comprehensive approach to modeling various behaviors potentially dictated by these equations in real-life scenarios.