Let $y=f(x)$ Be The Particular Solution To The Differential Equation $\frac{d Y}{d X}=\frac{1}{2 Y+1}$ With The Initial Condition $ Y ( 0 ) = 0 Y(0)=0 Y ( 0 ) = 0 [/tex]. Which Of The Following Gives An Expression For $f(x)$ And

Introduction

Differential equations are a fundamental concept in mathematics, describing how quantities change over time or space. In this article, we will delve into the world of separable differential equations, focusing on the particular solution to the equation $\frac{d y}{d x}=\frac{1}{2 y+1}$ with the initial condition $y(0)=0$. We will explore the concept of separable differential equations, the method of separation of variables, and the solution to the given differential equation.

What are Separable Differential Equations?

A separable differential equation is a type of differential equation that can be written in the form $\frac{dy}{dx} = f(x)g(y)$, where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$. The key characteristic of separable differential equations is that they can be separated into two distinct functions, one depending on $x$ and the other on $y$. This allows us to solve the equation by integrating both sides separately.

The Method of Separation of Variables

The method of separation of variables is a powerful technique used to solve separable differential equations. The basic idea is to separate the variables $x$ and $y$ by dividing both sides of the equation by $g(y)$ and integrating the resulting expression. This method is based on the fact that the derivative of a product is equal to the derivative of the first function times the second function, plus the derivative of the second function times the first function.

Solving the Differential Equation

Now, let's apply the method of separation of variables to the given differential equation $\frac{d y}{d x}=\frac{1}{2 y+1}$. We can rewrite the equation as $\frac{dy}{dx} = \frac{1}{2y+1}$.

To separate the variables, we divide both sides of the equation by $g(y) = 2y+1$, which gives us:

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

Next, we integrate both sides of the equation:

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

Using the substitution $u = 2y+1$, we get:

$$\int \frac{du}{u} = \int dx$$

Evaluating the integrals, we get:

$$\ln|u| = x + C$$

Substituting back $u = 2y+1$, we get:

$$\ln|2y+1| = x + C$$

Finding the Particular Solution

We are given the initial condition $y(0)=0$. To find the particular solution, we substitute $x=0$ and $y=0$ into the general solution:

$$\ln|2(0)+1| = 0 + C$$

Simplifying, we get:

$$\ln|1| = C$$

Since $\ln|1| = 0$, we have:

$$C = 0$$

Substituting $C=0$ back into the general solution, we get:

$$\ln|2y+1| = x$$

Solving for $y$

To solve for $y$, we exponentiate both sides of the equation:

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

Simplifying, we get:

$$|2y+1| = e^x$$

Since $y(0)=0$, we know that $y$ is non-negative. Therefore, we can drop the absolute value sign:

$$2y+1 = e^x$$

Solving for $y$, we get:

$$y = \frac{e^x-1}{2}$$

Conclusion

In this article, we have explored the concept of separable differential equations, the method of separation of variables, and the solution to the differential equation $\frac{d y}{d x}=\frac{1}{2 y+1}$ with the initial condition $y(0)=0$. We have shown that the particular solution to the differential equation is $y = \frac{e^x-1}{2}$. This solution provides a clear understanding of how the quantity $y$ changes over time, given the initial condition.

Applications of Separable Differential Equations

Separable differential equations have numerous applications in various fields, including physics, engineering, and economics. Some examples of applications include:

• Population growth models: Separable differential equations can be used to model the growth of populations over time.
• Chemical reactions: Separable differential equations can be used to model the rates of chemical reactions.
• Economic models: Separable differential equations can be used to model the behavior of economic systems.

Future Directions

In conclusion, separable differential equations are a powerful tool for modeling and analyzing complex systems. As we continue to explore the world of mathematics, we will undoubtedly encounter more applications of separable differential equations. By understanding the method of separation of variables and the solution to the given differential equation, we can gain a deeper appreciation for the beauty and power of mathematics.

References

• [1]: "Differential Equations and Dynamical Systems" by Lawrence Perko
• [2]: "Introduction to Differential Equations" by James R. Brannan and William E. Boyce
• [3]: "Separable Differential Equations" by Wolfram MathWorld

Glossary

• Separable differential equation: A differential equation that can be written in the form $\frac{dy}{dx} = f(x)g(y)$.
• Method of separation of variables: A technique used to solve separable differential equations by separating the variables $x$ and $y$.
• Particular solution: A solution to a differential equation that satisfies a given initial condition.
• General solution: A solution to a differential equation that satisfies the equation for all values of the independent variable.
  Separable Differential Equations: A Comprehensive Guide ===========================================================

Q&A: Separable Differential Equations

Q: What is a separable differential equation?

A: A separable differential equation is a type of differential equation that can be written in the form $\frac{dy}{dx} = f(x)g(y)$, where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$.

Q: How do I know if a differential equation is separable?

A: To determine if a differential equation is separable, look for the following characteristics:

• The equation can be written in the form $\frac{dy}{dx} = f(x)g(y)$.
• The function $f(x)$ is a function of $x$ only.
• The function $g(y)$ is a function of $y$ only.

Q: What is the method of separation of variables?

A: The method of separation of variables is a technique used to solve separable differential equations by separating the variables $x$ and $y$. This involves dividing both sides of the equation by $g(y)$ and integrating the resulting expression.

Q: How do I apply the method of separation of variables?

A: To apply the method of separation of variables, follow these steps:

1. Write the differential equation in the form $\frac{dy}{dx} = f(x)g(y)$.
2. Divide both sides of the equation by $g(y)$.
3. Integrate both sides of the equation.
4. Solve for $y$.

Q: What is the difference between a particular solution and a general solution?

A: A particular solution is a solution to a differential equation that satisfies a given initial condition. A general solution is a solution to a differential equation that satisfies the equation for all values of the independent variable.

Q: How do I find the particular solution to a differential equation?

A: To find the particular solution to a differential equation, follow these steps:

1. Write the differential equation in the form $\frac{dy}{dx} = f(x)g(y)$.
2. Apply the method of separation of variables.
3. Substitute the initial condition into the general solution.
4. Solve for $y$.

Q: What are some common applications of separable differential equations?

A: Separable differential equations have numerous applications in various fields, including:

• Population growth models
• Chemical reactions
• Economic models

Q: How do I know if a differential equation is separable?

A: To determine if a differential equation is separable, look for the following characteristics:

• The equation can be written in the form $\frac{dy}{dx} = f(x)g(y)$.
• The function $f(x)$ is a function of $x$ only.
• The function $g(y)$ is a function of $y$ only.

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

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

• Not separating the variables correctly.
• Not integrating both sides of the equation.
• Not solving for $y$ correctly.

Q: How do I verify that my solution is correct?

A: To verify that your solution is correct, follow these steps:

1. Check that the solution satisfies the differential equation.
2. Check that the solution satisfies the initial condition.
3. Check that the solution is consistent with the physical or real-world context of the problem.

Conclusion

In this article, we have explored the concept of separable differential equations, the method of separation of variables, and the solution to the differential equation $\frac{d y}{d x}=\frac{1}{2 y+1}$ with the initial condition $y(0)=0$. We have also provided a Q&A section to help readers understand the concepts and techniques involved in solving separable differential equations. By following the steps outlined in this article, readers can gain a deeper understanding of separable differential equations and how to apply them to real-world problems.