Finding The General Solution Of X²y' + Xy = 6y A Comprehensive Guide
Introduction to Differential Equations
In the realm of mathematics, differential equations stand as powerful tools for modeling and understanding phenomena that change over time or space. These equations involve unknown functions and their derivatives, making them essential in fields like physics, engineering, economics, and biology. This article delves into the process of finding the general solution of a specific differential equation, providing a comprehensive understanding of the methods and concepts involved.
At its core, a differential equation expresses a relationship between a function and its derivatives. The order of a differential equation is determined by the highest derivative present in the equation. For instance, an equation involving only the first derivative is a first-order differential equation, while one involving the second derivative is a second-order differential equation, and so on.
Differential equations can be broadly classified into two main categories: ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of a single independent variable and their derivatives, while PDEs involve functions of multiple independent variables and their partial derivatives. The differential equation we will explore in this article, x²y' + xy = 6y, falls under the category of ODEs, as it involves a function y of a single variable x and its first derivative y'. Understanding these fundamental concepts sets the stage for tackling the problem at hand.
Problem Statement: x²y' + xy = 6y
Our primary objective is to find the general solution of the given differential equation: x²y' + xy = 6y. This equation is a first-order ordinary differential equation, which means it involves the first derivative of the unknown function y with respect to x. The general solution of a differential equation represents a family of functions that satisfy the equation. It typically includes arbitrary constants, which can be determined by applying initial conditions or boundary conditions, if provided.
Before diving into the solution process, let's take a closer look at the equation. We can rewrite it as:
x²(dy/dx) + xy = 6y
Here, y' is expressed as dy/dx, which represents the derivative of y with respect to x. This form highlights the relationship between the rate of change of y and the variables x and y themselves.
To solve this differential equation, we will employ a technique known as separation of variables. This method involves rearranging the equation so that terms involving y and dy are on one side, and terms involving x and dx are on the other. This separation allows us to integrate both sides independently, leading to the general solution. The separation of variables technique is a powerful tool for solving certain types of differential equations, particularly those that can be expressed in a separable form. It is a cornerstone method in the study of differential equations and has wide applications in various scientific and engineering disciplines.
Method: Separation of Variables
The separation of variables technique is a fundamental method for solving first-order ordinary differential equations. This approach is applicable when the equation can be rearranged such that the terms involving the dependent variable (y) and its differential (dy) are on one side, while the terms involving the independent variable (x) and its differential (dx) are on the other side. The power of this method lies in its ability to transform a differential equation into two separate integrals, which can then be solved independently.
To apply the separation of variables method to our equation, x²y' + xy = 6y, we first need to rewrite the equation in a form suitable for separation. Recall that y' represents dy/dx, so we can rewrite the equation as:
x²(dy/dx) + xy = 6y
Now, our goal is to isolate the terms involving y and dy on one side and the terms involving x and dx on the other. To achieve this, we can follow these steps:
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Subtract xy from both sides:
x²(dy/dx) = 6y - xy
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Factor out y on the right side:
x²(dy/dx) = y(6 - x)
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Divide both sides by x²:
dy/dx = (y(6 - x))/x²
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Divide both sides by y:
(1/y)(dy/dx) = (6 - x)/x²
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Multiply both sides by dx:
(1/y) dy = (6 - x)/x² dx
Now, the equation is successfully separated. All the terms involving y and dy are on the left side, and all the terms involving x and dx are on the right side. This separation allows us to integrate both sides independently, which is the next step in finding the general solution.
Solving the Separated Equation
Having successfully separated the variables in the differential equation, we now have:
(1/y) dy = (6 - x)/x² dx
The next crucial step is to integrate both sides of the equation. This will allow us to find the relationship between y and x and ultimately determine the general solution. Let's begin by integrating each side separately.
Integrating the Left Side
The left side of the equation is (1/y) dy. The integral of 1/y with respect to y is a standard integral, which is the natural logarithm of the absolute value of y. Thus, we have:
∫(1/y) dy = ln|y| + C₁
Here, C₁ represents the constant of integration. It's essential to include the constant of integration when evaluating indefinite integrals, as it accounts for the family of functions that have the same derivative.
Integrating the Right Side
The right side of the equation is (6 - x)/x² dx. To integrate this, we first need to simplify the expression by dividing each term in the numerator by x²:
(6 - x)/x² = 6/x² - x/x² = 6x⁻² - 1/x
Now, we can integrate each term separately:
∫(6x⁻² - 1/x) dx = ∫6x⁻² dx - ∫(1/x) dx
The integral of 6x⁻² with respect to x is:
∫6x⁻² dx = 6∫x⁻² dx = 6(-x⁻¹) + C₂ = -6/x + C₂
The integral of 1/x with respect to x is the natural logarithm of the absolute value of x:
∫(1/x) dx = ln|x| + C₃
Combining these results, we get:
∫(6x⁻² - 1/x) dx = -6/x - ln|x| + C₂ + C₃
We can combine the constants of integration C₂ and C₃ into a single constant, say C₄:
∫(6x⁻² - 1/x) dx = -6/x - ln|x| + C₄
Combining the Results
Now that we have integrated both sides of the equation, we can combine the results:
ln|y| + C₁ = -6/x - ln|x| + C₄
We can combine the constants C₁ and C₄ into a single constant, say C:
ln|y| = -6/x - ln|x| + C
This equation represents the implicit solution of the differential equation. To find the explicit solution, we need to isolate y.
Finding the General Solution
Having integrated both sides of the separated differential equation, we arrived at the implicit solution:
ln|y| = -6/x - ln|x| + C
To find the general solution, we need to express y explicitly as a function of x. This involves isolating y on one side of the equation.
Exponentiating Both Sides
The key to isolating y is to exponentiate both sides of the equation using the exponential function e (the base of the natural logarithm). This will effectively undo the natural logarithm on the left side. Applying the exponential function to both sides, we get:
e^(ln|y|) = e^(-6/x - ln|x| + C)
Using the property that e^(ln(a)) = a, the left side simplifies to:
|y| = e^(-6/x - ln|x| + C)
Simplifying the Right Side
The right side can be simplified using the properties of exponents. Recall that e^(a + b) = e^a * e^b. Applying this property, we can rewrite the right side as:
e^(-6/x - ln|x| + C) = e^(-6/x) * e^(-ln|x|) * e^C
We can further simplify e^(-ln|x|) using the property that e^(-ln(a)) = 1/a:
e^(-ln|x|) = 1/e^(ln|x|) = 1/|x|
Also, e^C is a constant, so we can replace it with another constant, say A:
e^C = A
Substituting these simplifications back into the equation, we get:
|y| = e^(-6/x) * (1/|x|) * A
Removing the Absolute Value
To remove the absolute value signs, we can consider both positive and negative values of y. This leads to:
y = ± A * e^(-6/x) / |x|
Since A is an arbitrary constant, ±A is also an arbitrary constant. We can replace ±A with a new constant, say K:
y = K * e^(-6/x) / |x|
Finally, we can rewrite |x| as x if we allow K to take on both positive and negative values. This gives us the general solution:
y = K * e^(-6/x) / x
This is the general solution of the given differential equation. It represents a family of functions that satisfy the equation. The constant K can be determined if we are given an initial condition or a boundary condition.
Conclusion
In this article, we successfully found the general solution of the differential equation x²y' + xy = 6y. We employed the method of separation of variables, a powerful technique for solving first-order ordinary differential equations. This method involves separating the variables x and y and their differentials, integrating both sides of the equation, and then solving for y to obtain the general solution.
We began by rewriting the equation in a separable form and then integrated both sides, obtaining an implicit solution. To find the explicit general solution, we exponentiated both sides, simplified the expression using properties of exponents and logarithms, and ultimately arrived at:
y = K * e^(-6/x) / x
This solution represents a family of functions, each characterized by a different value of the arbitrary constant K. The value of K can be determined if we are given additional information, such as an initial condition or a boundary condition.
The process of solving differential equations is a fundamental aspect of mathematics and has wide-ranging applications in various fields of science and engineering. Understanding the techniques and concepts involved is crucial for modeling and analyzing dynamic systems. The separation of variables method, as demonstrated in this article, is a valuable tool in the arsenal of any mathematician or scientist dealing with differential equations.
By mastering these techniques, one can gain a deeper appreciation for the power and elegance of differential equations in describing the world around us. Whether it's modeling population growth, analyzing the motion of objects, or understanding the behavior of electrical circuits, differential equations provide the framework for understanding and predicting change.
