Solving (D^3 - 7D^2 + 14D - 8)y = E^x Cos 2x A Step-by-Step Guide
Introduction to Differential Equations
In the realm of mathematics, differential equations stand as powerful tools for modeling various phenomena across diverse fields, including physics, engineering, economics, and biology. These equations express relationships between a function and its derivatives, capturing the essence of change and rates of change. Solving a differential equation involves finding the function that satisfies the given equation. This article delves into the process of solving a specific type of differential equation, a third-order linear ordinary differential equation with constant coefficients, further complicated by a non-homogeneous term involving an exponential function and a cosine function. Understanding the methods to tackle such equations is crucial for anyone venturing into advanced mathematical modeling and analysis. The equation we aim to solve is (D^3 - 7D^2 + 14D - 8)y = e^x cos 2x, where 'D' represents the differential operator d/dx, and 'y' is the function we seek to determine. This particular equation is a testament to the intricacies of differential equations, demanding a blend of techniques to arrive at a solution. The journey to solve this equation will not only illuminate the methods involved but also underscore the broader applicability of differential equations in understanding the world around us. From the oscillations of a pendulum to the spread of a disease, differential equations provide the language to describe and predict dynamic systems.
Understanding the Differential Operator and Auxiliary Equation
To effectively solve the given differential equation, (D^3 - 7D^2 + 14D - 8)y = e^x cos 2x, it's crucial to first understand the differential operator 'D' and how it transforms our equation. The differential operator D is a symbolic representation of differentiation with respect to the independent variable, in this case, x. Thus, Dy means dy/dx, D^2y means d2y/dx2, and so forth. The polynomial in D, namely (D^3 - 7D^2 + 14D - 8), acts as an operator on the function y(x), transforming it into a linear combination of its derivatives. We first address the homogeneous part of the differential equation, which is (D^3 - 7D^2 + 14D - 8)y = 0. This leads us to the auxiliary equation, a critical step in finding the general solution. The auxiliary equation is obtained by replacing the differential operator D with an algebraic variable, typically denoted by 'm'. Thus, our auxiliary equation becomes m^3 - 7m^2 + 14m - 8 = 0. Solving this cubic equation is paramount as its roots dictate the form of the homogeneous solution. The roots of the auxiliary equation can be real and distinct, real and repeated, or complex conjugates. Each case necessitates a different approach in constructing the general solution. For instance, distinct real roots lead to exponential terms, repeated roots to polynomial-exponential terms, and complex conjugate roots to sinusoidal terms. Finding these roots is not merely an algebraic exercise; it's a gateway to understanding the fundamental behaviors inherent in the differential equation. The roots encapsulate the natural frequencies and damping characteristics of the system being modeled, offering insights into its stability and response to external influences. Therefore, a thorough grasp of how to derive and interpret the auxiliary equation is indispensable in the broader context of differential equation analysis.
Solving the Auxiliary Equation and Finding the Complementary Function
The process of finding the roots of the auxiliary equation m^3 - 7m^2 + 14m - 8 = 0 is a critical step in solving the differential equation. This cubic equation can be solved using various methods, such as factoring, synthetic division, or numerical techniques. By observation, we can see that m = 1 is a root since 1^3 - 7(1)^2 + 14(1) - 8 = 0. Using synthetic division or polynomial long division, we can then divide the cubic polynomial by (m - 1) to obtain a quadratic factor. This yields (m - 1)(m^2 - 6m + 8) = 0. The quadratic factor can be further factored as (m - 2)(m - 4) = 0. Thus, the roots of the auxiliary equation are m1 = 1, m2 = 2, and m3 = 4. Since we have three distinct real roots, the complementary function, which is the general solution to the homogeneous equation, takes the form y_c(x) = C_1e^{x} + C_2e^{2x} + C_3e^{4x}, where C_1, C_2, and C_3 are arbitrary constants. The complementary function represents the natural response of the system described by the differential equation, independent of the non-homogeneous term. It is a superposition of exponential functions, each corresponding to a root of the auxiliary equation. The arbitrary constants reflect the initial conditions or boundary conditions that would be needed to obtain a unique solution. Understanding the complementary function is vital as it forms a key component of the general solution to the non-homogeneous differential equation. It captures the inherent behavior of the system, such as its stability and oscillatory tendencies, before considering external influences. In essence, the complementary function provides a baseline understanding of the system's dynamics, setting the stage for analyzing the effects of the non-homogeneous term.
Determining the Particular Integral
To solve the non-homogeneous differential equation (D^3 - 7D^2 + 14D - 8)y = e^x cos 2x completely, we must now find the particular integral, y_p(x). The particular integral is a specific solution to the non-homogeneous equation and accounts for the effect of the non-homogeneous term, e^x cos 2x, on the system. The method of undetermined coefficients is a common technique for finding the particular integral when the non-homogeneous term has a specific form, such as exponential, sinusoidal, or polynomial functions, or combinations thereof. In our case, the non-homogeneous term is e^x cos 2x, which is a product of an exponential function and a cosine function. Therefore, we assume a particular integral of the form y_p(x) = e^x(A cos 2x + B sin 2x), where A and B are constants to be determined. This form is chosen because it includes both the cosine and sine functions multiplied by the exponential function, accounting for all possible terms that could arise when differentiating e^x cos 2x. The next step involves differentiating y_p(x) three times, using the product rule and chain rule as necessary, and then substituting these derivatives into the original differential equation. This will result in an equation involving e^x cos 2x and e^x sin 2x, with coefficients that are linear combinations of A and B. By equating the coefficients of e^x cos 2x and e^x sin 2x on both sides of the equation, we obtain a system of linear equations in A and B. Solving this system will give us the values of A and B, which then define the particular integral y_p(x). The particular integral represents the forced response of the system to the external input described by the non-homogeneous term. It shows how the system's behavior is directly influenced by the driving force. Finding the particular integral is a crucial step in obtaining the complete solution to the non-homogeneous differential equation, as it captures the specific response to the external stimulus.
Calculating Derivatives and Substituting into the Equation
To proceed with the method of undetermined coefficients, we must calculate the derivatives of our assumed particular integral, y_p(x) = e^x(A cos 2x + B sin 2x). The first derivative, y_p'(x), is found using the product rule: y_p'(x) = e^x(A cos 2x + B sin 2x) + e^x(-2A sin 2x + 2B cos 2x) = e^x[(A + 2B) cos 2x + (B - 2A) sin 2x]. The second derivative, y_p''(x), is calculated similarly: y_p''(x) = e^x[(A + 2B) cos 2x + (B - 2A) sin 2x] + e^x[-2(A + 2B) sin 2x + 2(B - 2A) cos 2x] = e^x[(-3A + 4B) cos 2x + (-4A - 3B) sin 2x]. Finally, the third derivative, y_p'''(x), is: y_p'''(x) = e^x[(-3A + 4B) cos 2x + (-4A - 3B) sin 2x] + e^x[-2(-3A + 4B) sin 2x + 2(-4A - 3B) cos 2x] = e^x[(-11A - 2B) cos 2x + (2A - 11B) sin 2x]. Now, we substitute y_p(x), y_p'(x), y_p''(x), and y_p'''(x) into the original differential equation, (D^3 - 7D^2 + 14D - 8)y = e^x cos 2x. This gives us: e^x[(-11A - 2B) cos 2x + (2A - 11B) sin 2x] - 7e^x[(-3A + 4B) cos 2x + (-4A - 3B) sin 2x] + 14e^x[(A + 2B) cos 2x + (B - 2A) sin 2x] - 8e^x[A cos 2x + B sin 2x] = e^x cos 2x. This substitution is a crucial step as it sets up the equation for determining the unknown coefficients A and B. The complexity of these calculations underscores the importance of careful and methodical differentiation. The resulting equation now contains terms involving e^x cos 2x and e^x sin 2x, which we can use to form a system of linear equations.
Solving for Coefficients and Obtaining the Particular Integral
After substituting the derivatives into the differential equation, we gather terms and equate the coefficients of e^x cos 2x and e^x sin 2x on both sides. This yields two equations: For e^x cos 2x: (-11A - 2B) - 7(-3A + 4B) + 14(A + 2B) - 8A = 1, which simplifies to 12A - 2B = 1. For e^x sin 2x: (2A - 11B) - 7(-4A - 3B) + 14(B - 2A) - 8B = 0, which simplifies to -2A - 12B = 0. We now have a system of two linear equations in two unknowns, A and B: 12A - 2B = 1 -2A - 12B = 0 Solving this system, we can multiply the second equation by 6 to get -12A - 72B = 0. Adding this to the first equation, we eliminate A and get -74B = 1, so B = -1/74. Substituting B back into the second equation, -2A - 12(-1/74) = 0, gives -2A + 12/74 = 0, so 2A = 12/74, and A = 6/74 = 3/37. Thus, we have found the coefficients A = 3/37 and B = -1/74. Now we can write the particular integral: y_p(x) = e^x[(3/37) cos 2x - (1/74) sin 2x]. This particular integral is a specific solution to the non-homogeneous differential equation, representing the system's response to the external force e^x cos 2x. The values of A and B determine the amplitude and phase of this response. Finding these coefficients is a critical step in completing the solution, as it quantifies the influence of the non-homogeneous term on the system's behavior. With the particular integral in hand, we can now combine it with the complementary function to obtain the general solution.
Forming the General Solution
Having found both the complementary function and the particular integral, we can now construct the general solution to the non-homogeneous differential equation (D^3 - 7D^2 + 14D - 8)y = e^x cos 2x. The general solution is the sum of the complementary function, y_c(x), and the particular integral, y_p(x): y(x) = y_c(x) + y_p(x). We previously found the complementary function to be y_c(x) = C_1e^x + C_2e^2x} + C_3e^{4x}, where C_1, C_2, and C_3 are arbitrary constants, and the particular integral to be y_p(x) = e^x[(3/37) cos 2x - (1/74) sin 2x]. Therefore, the general solution is + C_3e^{4x} + e^x[(3/37) cos 2x - (1/74) sin 2x]. This general solution represents the family of all possible solutions to the differential equation. The arbitrary constants C_1, C_2, and C_3 allow for a range of solutions, each corresponding to different initial conditions or boundary conditions. The complementary function describes the natural behavior of the system, while the particular integral describes the system's response to the external force e^x cos 2x. The general solution combines these two aspects, providing a complete picture of the system's dynamics. To find a unique solution, we would need to specify initial conditions, such as the value of y(x) and its derivatives at a particular point. These conditions would allow us to solve for the constants C_1, C_2, and C_3, thereby pinning down a specific solution curve. The general solution, however, provides a powerful framework for understanding the range of possible behaviors of the system described by the differential equation.
Conclusion: Significance and Applications of Solving Differential Equations
In conclusion, we have successfully solved the third-order linear non-homogeneous differential equation (D^3 - 7D^2 + 14D - 8)y = e^x cos 2x. This involved finding the roots of the auxiliary equation, constructing the complementary function, determining the particular integral using the method of undetermined coefficients, and finally, forming the general solution by combining these components. This process illustrates the power and versatility of differential equations in modeling and solving problems across various disciplines. The ability to solve such equations is not merely an academic exercise; it has profound practical implications. Differential equations are the backbone of many scientific and engineering disciplines. They are used to model the motion of objects, the flow of fluids, the spread of diseases, the behavior of electrical circuits, and countless other phenomena. Understanding and solving differential equations allows us to predict and control these systems, leading to advancements in technology and our understanding of the natural world. The specific equation we tackled, while complex, is representative of many real-world systems that exhibit oscillatory behavior influenced by external forces. The solution we obtained provides insights into how such systems respond to these forces, which is crucial for designing and optimizing various applications. For instance, in electrical engineering, this type of equation could model the behavior of an RLC circuit driven by an AC source. In mechanical engineering, it could describe the vibrations of a damped oscillator subjected to a periodic force. The methods and concepts discussed in this article are therefore essential tools for any scientist or engineer seeking to analyze and design dynamic systems. The study of differential equations is a cornerstone of mathematical modeling, and its applications are virtually limitless, underscoring its significance in both theoretical and applied contexts.
