Solving Y'' - 2y' + 2y = E^x Sec X A Step By Step Guide
In this article, we will delve into the process of solving the second-order linear non-homogeneous differential equation:
y'' - 2y' + 2y = e^x \sec x
This type of equation appears frequently in various fields of science and engineering, including physics, electrical circuits, and control systems. We will systematically break down the solution process, starting with finding the characteristic equation and its roots, then determining the homogeneous solution, and finally, employing the method of variation of parameters to find the particular solution. By the end of this discussion, you will have a comprehensive understanding of how to solve such differential equations. Understanding these concepts is crucial for anyone working with dynamic systems, as differential equations are the backbone for modeling and predicting system behavior. Mastering differential equations allows for the accurate modeling and simulation of physical phenomena, providing a strong foundation for advanced studies and practical applications. The journey through the solution process not only enhances your mathematical skills but also sharpens your analytical thinking, a valuable asset in problem-solving scenarios.
1. Characteristic Equation
To begin, we find the characteristic equation associated with the homogeneous part of the differential equation. The homogeneous equation is:
y'' - 2y' + 2y = 0
We replace y'' with m², y' with m, and y with 1, yielding the characteristic equation:
m^2 - 2m + 2 = 0
This quadratic equation is fundamental to finding the solutions to the homogeneous differential equation. Solving this equation accurately is a critical first step. The roots of the characteristic equation dictate the form of the homogeneous solution. For example, real and distinct roots lead to exponential solutions, while complex roots lead to sinusoidal solutions. The characteristic equation is more than just a mathematical tool; it is a bridge connecting the algebraic world of equations to the dynamic behavior described by the differential equation. Analyzing the discriminant of the quadratic equation can provide insights into the nature of the solutions even before they are explicitly calculated. This step allows us to predict whether the system will exhibit oscillatory behavior, exponential decay, or a combination of both. Understanding the nature of the roots is pivotal in interpreting the system's long-term behavior, making the characteristic equation a powerful analytical tool in the study of differential equations.
2. Roots of the Characteristic Equation
We solve the quadratic equation using the quadratic formula:
m = ${ -b \pm \sqrt{b^2 - 4ac} }$ / 2a
where a = 1, b = -2, and c = 2. Plugging in these values, we get:
m = ${ 2 \pm \sqrt{(-2)^2 - 4(1)(2)} }$ / 2(1)
m = ${ 2 \pm \sqrt{4 - 8} }$ / 2
m = ${ 2 \pm \sqrt{-4} }$ / 2
m = ${ 2 \pm 2i }$ / 2
m = 1 \pm i
Thus, the roots are complex conjugates: m₁ = 1 + i and m₂ = 1 - i. These complex roots indicate that the solutions to the homogeneous equation will involve sinusoidal functions multiplied by an exponential term. The complex roots are key to understanding the oscillatory nature of the system. The real part of the root (in this case, 1) governs the exponential growth or decay, while the imaginary part (in this case, 1) determines the frequency of oscillation. This interplay between exponential and sinusoidal components is a hallmark of systems with complex roots. The fact that the roots are complex conjugates ensures that the resulting solutions are real-valued, which is essential for representing physical phenomena. Understanding the implications of complex roots is vital in many areas, including electrical engineering (analyzing AC circuits) and mechanical engineering (studying damped oscillations). The roots' values give immediate insight into the stability and dynamic characteristics of the system, making their calculation and interpretation a critical skill in the analysis of differential equations.
3. Homogeneous Solution
Since the roots are complex conjugates, the homogeneous solution y_h(x) takes the form:
y_h(x) = e^(αx) (C₁ cos(βx) + C₂ sin(βx))
where α is the real part of the root and β is the imaginary part. In our case, α = 1 and β = 1, so the homogeneous solution is:
y_h(x) = e^x (C₁ cos(x) + C₂ sin(x))
Here, C₁ and C₂ are arbitrary constants determined by initial conditions. The homogeneous solution represents the natural response of the system, i.e., its behavior without any external forcing function. The exponential term e^x indicates that the oscillations will grow in amplitude over time, while the sinusoidal terms cos(x) and sin(x) describe the oscillatory nature of the solution. The constants C₁ and C₂ allow the solution to fit specific initial conditions, which provide information about the state of the system at a particular time. The homogeneous solution is a crucial component of the complete solution, as it captures the inherent dynamics of the system. Understanding the homogeneous solution provides insights into the system's stability and its tendency to oscillate or decay. In many physical systems, the homogeneous solution represents the transient behavior, which eventually fades away as time progresses. However, in systems with complex roots and a positive real part, the homogeneous solution can lead to instability, where oscillations grow without bound. Therefore, a thorough understanding of the homogeneous solution is essential for analyzing and controlling dynamic systems.
4. Particular Solution using Variation of Parameters
To find the particular solution y_p(x), we use the method of variation of parameters. This method is particularly useful for non-homogeneous equations where the forcing function (the term on the right-hand side of the equation) does not have a simple form. In our case, the forcing function is e^x sec(x). We assume a particular solution of the form:
y_p(x) = u₁(x) cos(x)e^x + u₂(x) sin(x)e^x
where u₁(x) and u₂(x) are functions to be determined. The method of variation of parameters involves several steps, including setting up a system of equations and solving for the derivatives of the unknown functions. This method is a powerful technique for finding particular solutions to non-homogeneous differential equations. It is especially valuable when the method of undetermined coefficients is not applicable, such as when the forcing function is not a polynomial, exponential, or sinusoidal function, or a combination thereof. Variation of parameters leverages the homogeneous solutions to construct a particular solution, making it a versatile tool. The process involves finding the Wronskian, a determinant that captures the linear independence of the homogeneous solutions. The Wronskian plays a crucial role in determining the unique solutions for the derivatives of the unknown functions. While the method can be computationally intensive, it provides a systematic approach to solving a wide range of non-homogeneous equations. Understanding the underlying principles of variation of parameters is essential for advanced studies in differential equations and their applications in various fields.
4.1 Setting up the equations
We need to find u₁'(x) and u₂'(x) such that:
u₁'(x) cos(x)e^x + u₂'(x) sin(x)e^x = 0
u₁'(x) (cos(x)e^x)' + u₂'(x) (sin(x)e^x)' = e^x sec(x)
Let y₁ = e^x cos(x) and y₂ = e^x sin(x). Then:
y₁' = e^x cos(x) - e^x sin(x)
y₂' = e^x sin(x) + e^x cos(x)
Our system of equations becomes:
u₁'(x) e^x cos(x) + u₂'(x) e^x sin(x) = 0
u₁'(x) (e^x cos(x) - e^x sin(x)) + u₂'(x) (e^x sin(x) + e^x cos(x)) = e^x sec(x)
Setting up these equations is a crucial step in the variation of parameters method. The first equation ensures that the assumed form of the particular solution satisfies the original differential equation, while the second equation incorporates the non-homogeneous term. The derivatives y₁' and y₂' are found using the product rule, as y₁ and y₂ are products of exponential and trigonometric functions. This system of equations can be solved using various methods, such as substitution or Cramer's rule. The key is to express u₁'(x) and u₂'(x) in terms of known functions, which can then be integrated to find u₁(x) and u₂(x). The structure of these equations reflects the fundamental principles of linear algebra and differential equations. The coefficients of u₁'(x) and u₂'(x) are derived from the homogeneous solutions, while the right-hand side is determined by the non-homogeneous term. The careful setup of these equations is essential for the successful application of the variation of parameters method.
4.2 Solving for u₁'(x) and u₂'(x)
We can solve this system of equations. First, divide the equations by e^x:
u₁'(x) cos(x) + u₂'(x) sin(x) = 0
u₁'(x) (cos(x) - sin(x)) + u₂'(x) (sin(x) + cos(x)) = sec(x)
From the first equation, we have:
u₁'(x) = -u₂'(x) (sin(x) / cos(x)) = -u₂'(x) tan(x)
Substitute this into the second equation:
-u₂'(x) tan(x) (cos(x) - sin(x)) + u₂'(x) (sin(x) + cos(x)) = sec(x)
Simplify:
-u₂'(x) (sin(x) - sin²(x) / cos(x)) + u₂'(x) (sin(x) + cos(x)) = sec(x)
u₂'(x) (-sin(x) + sin²(x) / cos(x) + sin(x) + cos(x)) = sec(x)
u₂'(x) (sin²(x) / cos(x) + cos(x)) = sec(x)
u₂'(x) (sin²(x) + cos²(x)) / cos(x) = sec(x)
u₂'(x) (1 / cos(x)) = sec(x)
u₂'(x) sec(x) = sec(x)
Thus:
u₂'(x) = 1
Now, substitute back to find u₁'(x):
u₁'(x) = -1 * tan(x)
Solving for u₁'(x) and u₂'(x) involves algebraic manipulation and trigonometric identities. The key is to eliminate one of the unknowns, allowing us to solve for the other. In this case, we express u₁'(x) in terms of u₂'(x) using the first equation and then substitute it into the second equation. This process simplifies the second equation, making it possible to solve for u₂'(x) directly. Once u₂'(x) is found, we can easily find u₁'(x) using the relationship derived from the first equation. The trigonometric identities, such as sin²(x) + cos²(x) = 1, play a crucial role in simplifying the equations. The resulting expressions for u₁'(x) and u₂'(x) are relatively simple, making the next step of integration straightforward. The ability to solve such systems of equations is a fundamental skill in mathematics and is essential for applying the variation of parameters method effectively. This step showcases the interplay between algebra, trigonometry, and differential equations, highlighting the interconnectedness of mathematical concepts.
4.3 Integrating to find u₁(x) and u₂(x)
Integrate u₁'(x) and u₂'(x) to find u₁(x) and u₂(x):
u₁(x) = ∫ -tan(x) dx = ln|cos(x)|
u₂(x) = ∫ 1 dx = x
Integrating to find u₁(x) and u₂(x) is a critical step in determining the particular solution. The integrals of u₁'(x) and u₂'(x) provide the functions that, when multiplied by the homogeneous solutions, will form the particular solution. The integral of -tan(x) is a standard result that can be found in integral tables or derived using substitution. The integral of 1 is simply x. These integration steps highlight the importance of calculus in solving differential equations. The constants of integration are typically omitted at this stage, as they would be absorbed into the constants in the homogeneous solution. The resulting functions u₁(x) and u₂(x) are essential components of the particular solution. Their forms reflect the nature of the forcing function and the homogeneous solutions. The ability to perform these integrations accurately is crucial for the successful application of the variation of parameters method. This step showcases the power of calculus in solving complex problems in differential equations.
4.4 Forming the Particular Solution
Now, substitute u₁(x) and u₂(x) into the expression for y_p(x):
y_p(x) = ln|cos(x)| e^x cos(x) + x e^x sin(x)
Forming the particular solution involves combining the functions u₁(x) and u₂(x) with the homogeneous solutions. This step synthesizes the results of the previous steps to create a solution that satisfies the non-homogeneous differential equation. The particular solution captures the response of the system to the specific forcing function. In this case, the particular solution consists of terms involving ln|cos(x)|, x, e^x, cos(x), and sin(x). The logarithmic term arises from the integral of -tan(x), while the linear term x comes from the integral of 1. The exponential and trigonometric terms are inherited from the homogeneous solutions. The particular solution adds a specific response to the system, complementing the natural response captured by the homogeneous solution. This step demonstrates the elegance of the variation of parameters method, where the particular solution is constructed from the homogeneous solutions and the forcing function. The particular solution is a key component of the complete solution and is essential for understanding the system's behavior under external influences.
5. General Solution
The general solution y(x) is the sum of the homogeneous solution y_h(x) and the particular solution y_p(x):
y(x) = y_h(x) + y_p(x)
y(x) = e^x (C₁ cos(x) + C₂ sin(x)) + ln|cos(x)| e^x cos(x) + x e^x sin(x)
This is the complete solution to the given differential equation. The general solution encompasses all possible solutions to the differential equation. It is the sum of the homogeneous solution, which captures the natural behavior of the system, and the particular solution, which accounts for the specific forcing function. The constants C₁ and C₂ in the homogeneous solution allow the general solution to satisfy specific initial conditions. The particular solution adds the necessary terms to address the non-homogeneous part of the equation. The general solution provides a comprehensive description of the system's behavior, including both its natural response and its response to external influences. This solution is a powerful tool for predicting the system's state at any given time. The process of finding the general solution involves a combination of techniques, including finding the characteristic equation, solving for its roots, determining the homogeneous solution, and employing a method like variation of parameters to find the particular solution. The general solution represents the culmination of these efforts and provides a complete understanding of the system's dynamics.
In this article, we have successfully solved the differential equation:
y'' - 2y' + 2y = e^x sec(x)
by finding the characteristic equation, determining its roots, constructing the homogeneous solution, and using the method of variation of parameters to find the particular solution. The general solution is:
y(x) = e^x (C₁ cos(x) + C₂ sin(x)) + ln|cos(x)| e^x cos(x) + x e^x sin(x)
This process demonstrates the power and elegance of differential equation techniques in solving complex problems. Solving differential equations is a cornerstone of mathematical modeling and analysis. The ability to find general solutions to these equations allows for the prediction and understanding of a wide range of phenomena in science and engineering. The techniques discussed in this article, such as finding characteristic equations and using variation of parameters, are fundamental tools in this endeavor. Differential equations provide a framework for describing how systems change over time and are essential for analyzing dynamic systems. The general solution represents the complete behavior of the system, accounting for both its natural tendencies and external influences. The constants in the homogeneous solution allow the general solution to be tailored to specific initial conditions, providing a precise description of the system's state. The process of solving differential equations not only enhances mathematical skills but also fosters critical thinking and problem-solving abilities. The applications of differential equations are vast, spanning diverse fields such as physics, engineering, biology, and economics. Mastering these techniques is crucial for anyone working with dynamic systems and seeking to understand and predict their behavior.
