Solve The Differential Equation:$\[ Y \frac{d Y}{d X} = \sec^2 X (2 \tan X + 1) \\]

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 $y \frac{dy}{dx} = \sec^2 x (2 \tan x + 1)$. This equation is a classic example of a separable differential equation, which can be solved using various techniques.

Understanding the Equation

Before we dive into the solution, let's understand the equation. The equation 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 given by:

$$y \frac{dy}{dx} = \sec^2 x (2 \tan x + 1)$$

Separating the Variables

To solve this equation, we need to separate the variables. This means we need to isolate the variables $y$ and $x$ on opposite sides of the equation. We can do this by dividing both sides of the equation by $y$:

$$\frac{dy}{dx} = \frac{\sec^2 x (2 \tan x + 1)}{y}$$

Integrating Both Sides

Now that we have separated the variables, we can integrate both sides of the equation. The left-hand side is the derivative of $y$ with respect to $x$, so we can integrate it directly:

$$\int \frac{dy}{dx} dx = \int \frac{\sec^2 x (2 \tan x + 1)}{y} dx$$

Using the fundamental theorem of calculus, we can evaluate the left-hand side as:

$$y = \int \frac{\sec^2 x (2 \tan x + 1)}{y} dx$$

Evaluating the Integral

Now we need to evaluate the integral on the right-hand side. We can do this by using the substitution method. Let's substitute $u = \tan x$, so that $du = \sec^2 x dx$. Then, we can rewrite the integral as:

$$\int \frac{\sec^2 x (2 \tan x + 1)}{y} dx = \int \frac{(2u + 1)}{y} du$$

Solving for y

Now that we have evaluated the integral, we can solve for $y$. We can do this by rearranging the equation:

$$y = \int \frac{(2u + 1)}{y} du$$

Using the fundamental theorem of calculus, we can evaluate the integral as:

$$y = \frac{1}{2} \int (2u + 1) du$$

Evaluating the integral, we get:

$$y = \frac{1}{2} (u^2 + u) + C$$

Substituting back $u = \tan x$, we get:

$$y = \frac{1}{2} (\tan^2 x + \tan x) + C$$

Simplifying the Solution

We can simplify the solution by combining like terms:

$$y = \frac{1}{2} \tan^2 x + \frac{1}{2} \tan x + C$$

Using the trigonometric identity $\tan^2 x + 1 = \sec^2 x$, we can rewrite the solution as:

$$y = \frac{1}{2} (\sec^2 x - 1) + \frac{1}{2} \tan x + C$$

Conclusion

In this article, we have solved the differential equation $y \frac{dy}{dx} = \sec^2 x (2 \tan x + 1)$. We used the technique of separating the variables and integrating both sides of the equation to arrive at the solution. The final solution is given by:

$$y = \frac{1}{2} (\sec^2 x - 1) + \frac{1}{2} \tan x + C$$

This solution is a classic example of a separable differential equation, and it demonstrates the power of using mathematical techniques to solve complex problems.

Applications of the Solution

The solution to this differential equation has many applications in mathematics and physics. For example, it can be used to model the motion of a particle in a potential field, or to describe the behavior of a system in a state of equilibrium. The solution can also be used to derive the equations of motion for a particle in a central force field.

Future Work

In future work, we can explore other techniques for solving differential equations, such as the method of undetermined coefficients or the method of variation of parameters. We can also apply the solution to real-world problems, such as modeling the behavior of a population or the motion of a celestial body.

References

• [1] Boyce, W. E., & DiPrima, R. C. (2012). Elementary differential equations and boundary value problems. John Wiley & Sons.
• [2] Edwards, C. H., & Penney, D. E. (2015). Differential equations and boundary value problems: Computing and modeling. Pearson Education.
• [3] Simmons, G. F. (2015). Differential equations with applications and historical notes. McGraw-Hill Education.
  Solving the Differential Equation: A Q&A Guide =====================================================

Introduction

In our previous article, we solved the differential equation $y \frac{dy}{dx} = \sec^2 x (2 \tan x + 1)$. In this article, we will provide a Q&A guide to help you understand the solution and its applications.

Q: What is the main concept behind solving this differential equation?

A: The main concept behind solving this differential equation is the technique of separating the variables. This involves isolating the variables $y$ and $x$ on opposite sides of the equation, and then integrating both sides to arrive at the solution.

Q: What is the significance of the substitution $u = \tan x$?

A: The substitution $u = \tan x$ is used to simplify the integral on the right-hand side of the equation. By substituting $u$ for $\tan x$, we can rewrite the integral in terms of $u$, which makes it easier to evaluate.

Q: How do we evaluate the integral on the right-hand side of the equation?

A: We evaluate the integral on the right-hand side of the equation by using the substitution method. We substitute $u = \tan x$, and then rewrite the integral in terms of $u$. We can then evaluate the integral using the fundamental theorem of calculus.

Q: What is the final solution to the differential equation?

A: The final solution to the differential equation is given by:

$$y = \frac{1}{2} (\sec^2 x - 1) + \frac{1}{2} \tan x + C$$

Q: What are some applications of the solution to this differential equation?

A: The solution to this differential equation has many applications in mathematics and physics. For example, it can be used to model the motion of a particle in a potential field, or to describe the behavior of a system in a state of equilibrium. The solution can also be used to derive the equations of motion for a particle in a central force field.

Q: How do we use the solution to model real-world problems?

A: We can use the solution to model real-world problems by applying the principles of differential equations to the problem at hand. For example, we can use the solution to model the behavior of a population, or the motion of a celestial body.

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

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

• Not separating the variables correctly
• Not using the correct substitution method
• Not evaluating the integral correctly
• Not applying the fundamental theorem of calculus correctly

Q: How do we check the validity of the solution?

A: We can check the validity of the solution by applying the initial conditions and boundary conditions to the solution. We can also use numerical methods to verify the solution.

Q: What are some advanced techniques for solving differential equations?

A: Some advanced techniques for solving differential equations include:

• The method of undetermined coefficients
• The method of variation of parameters
• The use of Laplace transforms
• The use of Fourier transforms

Conclusion

In this article, we have provided a Q&A guide to help you understand the solution to the differential equation $y \frac{dy}{dx} = \sec^2 x (2 \tan x + 1)$. We have discussed the main concept behind solving this differential equation, the significance of the substitution $u = \tan x$, and the final solution to the differential equation. We have also discussed some applications of the solution, and some common mistakes to avoid when solving differential equations.