Solving (D+1)^2 Y = X^2 E^{-x} Cos X A Step-by-Step Guide

This article delves into the intricacies of solving the second-order linear non-homogeneous differential equation (D+1)^2 y = x^2 e^{-x} cos x. This type of equation frequently appears in various fields of science and engineering, including physics, electrical circuits, and control systems. Understanding the methods to solve such equations is crucial for modeling and analyzing these systems. We will explore the step-by-step approach, highlighting the underlying concepts and techniques involved. This comprehensive guide aims to provide a clear and concise understanding of the solution process, suitable for students and professionals alike.

1. Understanding the Differential Equation

The given differential equation is (D+1)^2 y = x^2 e^{-x} cos x. This equation falls under the category of second-order linear non-homogeneous differential equations with constant coefficients. Let's break down the components to understand the equation better:

• Second-Order: The highest derivative present in the equation is the second derivative (represented implicitly by the (D+1)^2 operator), indicating the order of the differential equation.
• Linear: The equation is linear because the dependent variable y and its derivatives appear only to the first power and are not multiplied together. There are no terms like y^2 or y y'. The coefficients of y and its derivatives are functions of the independent variable x only.
• Non-Homogeneous: The presence of the term x^2 e^-x* cos x on the right-hand side makes the equation non-homogeneous. If the right-hand side were zero, the equation would be homogeneous.
• Constant Coefficients: The coefficients of the derivatives of y are constants. In this case, expanding (D+1)^2 gives us D^2 + 2D + 1, where the coefficients 1, 2, and 1 are constants.

To effectively solve this equation, we will first find the complementary function (the solution to the homogeneous equation) and then determine the particular integral (a solution that satisfies the non-homogeneous equation). The general solution will be the sum of the complementary function and the particular integral. This step-by-step approach ensures a structured and accurate solution.

2. Finding the Complementary Function

The first step in solving the differential equation (D+1)^2 y = x^2 e^{-x} cos x is to find the complementary function. The complementary function is the general solution to the corresponding homogeneous equation, which is obtained by setting the right-hand side of the original equation to zero. Thus, we consider the equation:

(D+1)^2 y = 0

Here, D represents the differential operator d/dx. To solve this homogeneous equation, we assume a solution of the form y = e^mx*, where m is a constant. Substituting this into the homogeneous equation, we get:

(m+1)^2 e^{mx} = 0

Since e^mx* is never zero, we have the auxiliary equation:

(m+1)^2 = 0

This equation has a repeated root m = -1. For repeated roots, the complementary function takes the form:

y_c = (C_1 + C_2 x)e^{-x}

where C1 and C2 are arbitrary constants. This complementary function represents the general solution to the homogeneous part of the differential equation. It describes the natural behavior of the system without the influence of the external forcing function (x^2 e^-x* cos x). The presence of the e^-x* term indicates a decaying exponential behavior, while the (C1 + C2 x) part accounts for the repeated root in the auxiliary equation. Understanding the complementary function is crucial as it forms a vital part of the overall solution to the non-homogeneous equation.

3. Determining the Particular Integral

Now, let's move on to finding the particular integral (y_p) for the differential equation (D+1)^2 y = x^2 e^{-x} cos x. The particular integral is a solution that satisfies the non-homogeneous equation. Since the right-hand side of the equation is of the form x^2 e^-x* cos x, we will use the method of undetermined coefficients to find y_p. This method involves assuming a particular solution of a similar form to the non-homogeneous term and then determining the coefficients by substituting the assumed solution into the original equation.

Based on the form of the non-homogeneous term (x^2 e^-x* cos x), we assume a particular integral of the following form:

y_p = e^{-x} [(Ax^2 + Bx + C)cos x + (Dx^2 + Ex + F)sin x]

This form includes polynomial terms up to degree 2 multiplied by both cos x and sin x, as well as the e^-x* term. The constants A, B, C, D, E, and F are the undetermined coefficients that we need to find. The complexity of this assumed solution arises from the presence of the polynomial term x^2, the exponential term e^-x*, and the trigonometric term cos x. Each of these components necessitates including corresponding terms in the assumed particular integral.

To find these coefficients, we need to differentiate y_p twice (to apply the (D+1)^2 operator) and substitute the result back into the original differential equation. This process will generate a system of linear equations that we can solve for A, B, C, D, E, and F. The differentiation process can be quite lengthy and requires careful application of the product rule and chain rule. Once we have the values of the coefficients, we will have determined the particular integral y_p. The particular integral represents a specific response of the system to the external forcing function and is crucial for obtaining the complete solution to the non-homogeneous equation.

4. Calculating Derivatives and Substitution

This section involves the most computationally intensive part of solving the differential equation (D+1)^2 y = x^2 e^{-x} cos x: calculating the derivatives of the assumed particular integral and substituting them back into the equation. Recall that our assumed particular integral is:

y_p = e^{-x} [(Ax^2 + Bx + C)cos x + (Dx^2 + Ex + F)sin x]

We need to find the first and second derivatives of y_p with respect to x. This requires careful application of the product rule and chain rule. Let's denote the two terms inside the brackets as:

u(x) = (Ax^2 + Bx + C)cos x + (Dx^2 + Ex + F)sin x

Then, y_p = e^-x* u(x). Now we can find the first derivative y_p':

y_p' = -e^{-x} u(x) + e^{-x} u'(x)

Next, we need to find u'(x), which involves differentiating each term in u(x) using the product rule:

u'(x) = (2Ax + B)cos x - (Ax^2 + Bx + C)sin x + (2Dx + E)sin x + (Dx^2 + Ex + F)cos x

Now we can find the second derivative y_p'' by differentiating y_p':

y_p'' = e^{-x} u(x) - 2e^{-x} u'(x) + e^{-x} u''(x)

We also need to find u''(x), which involves differentiating u'(x):

u''(x) = 2Acos x - (2Ax + B)sin x - (2Ax + B)sin x - (Ax^2 + Bx + C)cos x + 2Dsin x + (2Dx + E)cos x + (2Dx + E)cos x - (Dx^2 + Ex + F)sin x

After obtaining y_p and its derivatives, we need to substitute them into the left-hand side of the original differential equation, which is (D+1)^2 y. Remember that (D+1)^2 y = (D^2 + 2D + 1) y = y'' + 2y' + y. Substituting y_p, y_p', and y_p'' into this expression will result in a complex algebraic expression. This expression must then be simplified and equated to the right-hand side of the original differential equation, which is x^2 e^-x* cos x. This step leads to a system of linear equations for the undetermined coefficients A, B, C, D, E, and F. The meticulous calculation and simplification in this section are crucial for obtaining the correct particular integral. Errors in differentiation or substitution will propagate through the rest of the solution process.

5. Solving for the Undetermined Coefficients

After substituting the particular integral (y_p) and its derivatives into the original differential equation (D+1)^2 y = x^2 e^{-x} cos x, and simplifying the resulting expression, we arrive at an equation of the form:

e^{-x} [P(x) cos x + Q(x) sin x] = x^2 e^{-x} cos x

where P(x) and Q(x) are polynomials in x with coefficients that are linear combinations of the undetermined coefficients A, B, C, D, E, and F. To solve for these coefficients, we equate the coefficients of corresponding terms on both sides of the equation. This means equating the coefficients of x^2 cos x, x cos x, cos x, x^2 sin x, x sin x, and sin x separately. This process yields a system of six linear equations in six unknowns (A, B, C, D, E, and F).

The system of equations will typically look something like this:

• Coefficient of x^2 cos x: a1A + b1B + c1C + d1D + e1E + f1F = 1
• Coefficient of x cos x: a2A + b2B + c2C + d2D + e2E + f2F = 0
• Coefficient of cos x: a3A + b3B + c3C + d3D + e3E + f3F = 0
• Coefficient of x^2 sin x: a4A + b4B + c4C + d4D + e4E + f4F = 0
• Coefficient of x sin x: a5A + b5B + c5C + d5D + e5E + f5F = 0
• Coefficient of sin x: a6A + b6B + c6C + d6D + e6E + f6F = 0

where a1, b1, c1, ..., f6 are constants obtained from the substitution and simplification steps. Solving this system of linear equations can be done using various methods, including:

• Gaussian elimination: A systematic method for solving systems of linear equations by transforming the system into an equivalent upper triangular form.
• Matrix inversion: If the coefficient matrix is invertible, the solution can be found by multiplying the inverse of the matrix by the constant vector.
• Software tools: Software packages like MATLAB, Mathematica, or Python libraries (e.g., NumPy, SciPy) can be used to solve the system of equations efficiently.

The process of solving for the undetermined coefficients is crucial as it determines the specific form of the particular integral. Accurate solutions for A, B, C, D, E, and F are essential for obtaining the correct particular integral and, consequently, the general solution to the differential equation. This step often requires careful attention to detail and may involve significant algebraic manipulation.

6. Constructing the General Solution

Once we have found both the complementary function (y_c) and the particular integral (y_p), we can construct the general solution (y) to the non-homogeneous differential equation (D+1)^2 y = x^2 e^{-x} cos x. The general solution is simply the sum of the complementary function and the particular integral:

y = y_c + y_p

Recall that the complementary function we found was:

y_c = (C_1 + C_2 x)e^{-x}

where C1 and C2 are arbitrary constants. These constants arise from the two linearly independent solutions to the homogeneous equation and represent the degrees of freedom in the general solution.

The particular integral (y_p) has a more complex form, which we determined by the method of undetermined coefficients. It will be an expression of the form:

y_p = e^{-x} [(Ax^2 + Bx + C)cos x + (Dx^2 + Ex + F)sin x]

where A, B, C, D, E, and F are the specific values we found by solving the system of linear equations in the previous section. Substituting these values into the expression for y_p gives us the particular solution that satisfies the non-homogeneous equation.

Therefore, the general solution to the differential equation is:

y = (C_1 + C_2 x)e^{-x} + e^{-x} [(Ax^2 + Bx + C)cos x + (Dx^2 + Ex + F)sin x]

This general solution represents the family of all possible solutions to the differential equation. It includes two arbitrary constants, C1 and C2, which can be determined if we are given initial conditions or boundary conditions. Initial conditions typically specify the values of y and its derivative at a particular point, while boundary conditions specify the values of y at two different points. Applying these conditions allows us to find unique values for C1 and C2, resulting in a unique solution that satisfies the given conditions.

In summary, the general solution consists of two parts: the complementary function, which captures the natural behavior of the system, and the particular integral, which represents the system's response to the external forcing function. Together, they provide a complete description of the solutions to the non-homogeneous differential equation.

7. Conclusion

Solving the differential equation (D+1)^2 y = x^2 e^{-x} cos x involves a multi-step process that combines understanding the nature of the equation, finding the complementary function, determining the particular integral, and constructing the general solution. This article has provided a detailed walkthrough of each of these steps, highlighting the key concepts and techniques involved. Let's recap the main points:

1. Understanding the Differential Equation: We identified the equation as a second-order linear non-homogeneous differential equation with constant coefficients.
2. Finding the Complementary Function: We solved the corresponding homogeneous equation by finding the roots of the auxiliary equation and constructing the complementary function based on these roots.
3. Determining the Particular Integral: We used the method of undetermined coefficients, assuming a particular solution of a similar form to the non-homogeneous term and setting up the equation.
4. Calculating Derivatives and Substitution: This step involved the complex and meticulous differentiation of the assumed particular integral and substitution into the original equation.
5. Solving for the Undetermined Coefficients: We equated coefficients of corresponding terms and solved the resulting system of linear equations to find the values of the undetermined coefficients.
6. Constructing the General Solution: We combined the complementary function and the particular integral to form the general solution, which represents the family of all possible solutions to the differential equation.

The process of solving such differential equations is crucial in many areas of science and engineering, allowing us to model and analyze various systems and phenomena. While the calculations can be lengthy and complex, the systematic approach outlined in this article provides a clear and effective method for obtaining solutions. Furthermore, understanding the underlying concepts, such as linearity, homogeneity, and the role of the complementary function and particular integral, is essential for applying these techniques to different types of differential equations and problems. Mastering these techniques provides a powerful tool for tackling a wide range of mathematical and real-world challenges.