Solve The Differential Equation: 2 X Y D Y D X = 1 + Y 2 2xy \frac{dy}{dx} = 1 + Y^2 2 X Y D X D Y ​ = 1 + Y 2

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Introduction

Differential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the differential equation $2xy \frac{dy}{dx} = 1 + y^2$. This equation is a classic example of a first-order differential equation, and it can be solved using various techniques.

Understanding the Equation

Before we dive into solving the equation, let's understand what it represents. The equation $2xy \frac{dy}{dx} = 1 + y^2$ is a first-order differential equation, which means it involves a derivative of a function with respect to a single variable. In this case, the variable is $x$, and the function is $y$. The equation is a product of two functions, $2xy$ and $\frac{dy}{dx}$, which is equal to $1 + y^2$.

Separating Variables

One of the most common techniques for solving differential equations is to separate the variables. This involves rearranging the equation so that all the terms involving $y$ are on one side, and all the terms involving $x$ are on the other side. In this case, we can separate the variables by dividing both sides of the equation by $2xy$.

$$\frac{dy}{dx} = \frac{1 + y^2}{2xy}$$

Integrating Both Sides

Now that we have separated the variables, we can integrate both sides of the equation. The left-hand side is a derivative of $y$ with respect to $x$, so we can integrate it directly. The right-hand side is a function of $y$ and $x$, so we need to use a substitution to integrate it.

Let's start by integrating the left-hand side:

$$\int \frac{dy}{dx} dx = \int \frac{1 + y^2}{2xy} dx$$

Using the substitution $u = y^2$, we can rewrite the right-hand side as:

$$\int \frac{1 + y^2}{2xy} dx = \int \frac{1 + u}{2x \sqrt{u}} dx$$

Now we can integrate both sides:

$$\int \frac{dy}{dx} dx = \int \frac{1 + u}{2x \sqrt{u}} dx$$

$$y = \int \frac{1 + u}{2x \sqrt{u}} dx$$

Evaluating the Integral

The integral on the right-hand side is a bit tricky to evaluate, but we can use a substitution to simplify it. Let's substitute $v = \sqrt{u}$, so that $dv = \frac{1}{2\sqrt{u}} du$. Then we can rewrite the integral as:

$$\int \frac{1 + u}{2x \sqrt{u}} dx = \int \frac{1 + v^2}{2xv} dv$$

Now we can integrate both sides:

$$\int \frac{1 + v^2}{2xv} dv = \int \frac{1}{2xv} dv + \int \frac{v}{2xv} dv$$

$$\int \frac{1 + v^2}{2xv} dv = \frac{1}{2x} \ln |v| + \frac{1}{2} \ln |v|$$

Simplifying the Result

Now that we have evaluated the integral, we can simplify the result. We can start by substituting back $v = \sqrt{u}$:

$$\frac{1}{2x} \ln |v| + \frac{1}{2} \ln |v| = \frac{1}{2x} \ln |\sqrt{u}| + \frac{1}{2} \ln |\sqrt{u}|$$

$$\frac{1}{2x} \ln |\sqrt{u}| + \frac{1}{2} \ln |\sqrt{u}| = \frac{1}{2x} \ln |u| + \frac{1}{2} \ln |u|$$

Now we can substitute back $u = y^2$:

$$\frac{1}{2x} \ln |u| + \frac{1}{2} \ln |u| = \frac{1}{2x} \ln |y^2| + \frac{1}{2} \ln |y^2|$$

$$\frac{1}{2x} \ln |y^2| + \frac{1}{2} \ln |y^2| = \frac{1}{2x} (2 \ln |y|) + \frac{1}{2} (2 \ln |y|)$$

Final Result

Now that we have simplified the result, we can write the final solution to the differential equation:

$$y = \frac{1}{2x} (2 \ln |y|) + \frac{1}{2} (2 \ln |y|)$$

$$y = \frac{1}{x} \ln |y| + \ln |y|$$

$$y = \ln |y| \left( \frac{1}{x} + 1 \right)$$

This is the final solution to the differential equation $2xy \frac{dy}{dx} = 1 + y^2$. We can see that the solution involves a logarithmic function, which is a common feature of many differential equations.

Conclusion

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Q: What is a differential equation?

A: A differential equation is a mathematical equation that involves an unknown function and its derivatives. It is a fundamental concept in mathematics, and it has numerous applications in physics, engineering, economics, and other fields.

Q: What is the difference between a first-order and a second-order differential equation?

A: A first-order differential equation involves a derivative of a function with respect to a single variable, while a second-order differential equation involves a second derivative of a function with respect to a single variable.

Q: How do I solve a differential equation?

A: There are several techniques for solving differential equations, including separation of variables, integration by substitution, and using a computer algebra system. The technique you use will depend on the type of differential equation you are solving.

Q: What is the purpose of separating variables in a differential equation?

A: Separating variables is a technique used to solve differential equations by rearranging the equation so that all the terms involving the unknown function are on one side, and all the terms involving the independent variable are on the other side.

Q: How do I integrate a differential equation?

A: Integrating a differential equation involves finding the antiderivative of the function on the right-hand side of the equation. This can be done using various techniques, including substitution, integration by parts, and using a computer algebra system.

Q: What is the significance of the constant of integration in a differential equation?

A: The constant of integration is a constant that is added to the solution of a differential equation to account for the fact that the solution is not unique. It is a fundamental concept in differential equations, and it has numerous applications in physics, engineering, and other fields.

Q: Can I use a computer algebra system to solve a differential equation?

A: Yes, you can use a computer algebra system to solve a differential equation. Many computer algebra systems, such as Mathematica and Maple, have built-in functions for solving differential equations.

Q: What are some common types of differential equations?

A: Some common types of differential equations include:

• First-order linear differential equations
• Second-order linear differential equations
• Nonlinear differential equations
• Partial differential equations

Q: How do I apply differential equations in real-world problems?

A: Differential equations have numerous applications in real-world problems, including:

• Modeling population growth and decline
• Describing the motion of objects under the influence of gravity
• Analyzing the behavior of electrical circuits
• Studying the spread of diseases

Q: What are some common mistakes to avoid when solving differential equations?

A: Some common mistakes to avoid when solving differential equations include:

• Failing to check the domain of the solution
• Failing to check the range of the solution
• Failing to account for the constant of integration
• Failing to use the correct technique for solving the differential equation

Conclusion

In this article, we have answered some frequently asked questions about solving differential equations. We have discussed the basics of differential equations, including the definition, types, and techniques for solving them. We have also discussed some common mistakes to avoid when solving differential equations. We hope that this article has provided a clear and concise guide to solving differential equations, and that it will be helpful to students and professionals alike.