What Is The Particular Solution To The Differential Equation $\frac{d Y}{d X}=2 \sin (x)(3-y$\] With The Initial Condition $y(\pi)=2$?Answer Attempt 2 Out Of 2:$y=e^{-2 \cos X-3}-3$

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Introduction

In this article, we will explore the concept of a particular solution to a differential equation. A differential equation is a mathematical equation that involves an unknown function and its derivatives. The particular solution to a differential equation is a specific solution that satisfies the equation and the given initial conditions. In this case, we will focus on finding the particular solution to the differential equation $\frac{d y}{d x}=2 \sin (x)(3-y)$ with the initial condition $y(\pi)=2$.

Understanding the Differential Equation

The given differential equation is $\frac{d y}{d x}=2 \sin (x)(3-y)$. This equation involves an unknown function $y$ and its derivative $\frac{d y}{d x}$. The equation is a first-order differential equation, meaning that it involves only the first derivative of the unknown function. The equation is also nonlinear, meaning that the unknown function and its derivative appear in a nonlinear manner.

Separating the Variables

To solve the differential equation, we can use the method of separation of variables. This method involves separating the unknown function and its derivative into different parts of the equation. In this case, we can separate the variables as follows:

$$\frac{d y}{d x}=2 \sin (x)(3-y)$$

$$\frac{d y}{2 \sin (x)(3-y)}=d x$$

Integrating Both Sides

Now that we have separated the variables, we can integrate both sides of the equation. The left-hand side of the equation involves the unknown function $y$, while the right-hand side involves the variable $x$. We can integrate both sides as follows:

$$\int \frac{d y}{2 \sin (x)(3-y)}=\int d x$$

Using Partial Fractions

To integrate the left-hand side of the equation, we can use partial fractions. This involves expressing the left-hand side as a sum of simpler fractions. In this case, we can express the left-hand side as follows:

$$\frac{1}{2 \sin (x)(3-y)}=\frac{A}{\sin (x)}+\frac{B}{3-y}$$

Finding the Particular Solution

Now that we have expressed the left-hand side of the equation in terms of partial fractions, we can integrate both sides of the equation. The left-hand side involves the unknown function $y$, while the right-hand side involves the variable $x$. We can integrate both sides as follows:

$$\int \left(\frac{A}{\sin (x)}+\frac{B}{3-y}\right) d x=\int d x$$

Applying the Initial Condition

To find the particular solution, we need to apply the initial condition $y(\pi)=2$. This involves substituting the value of $y$ at $x=\pi$ into the general solution. We can substitute the value of $y$ as follows:

$$y(\pi)=e^{-2 \cos \pi-3}-3=2$$

Solving for the Particular Solution

Now that we have applied the initial condition, we can solve for the particular solution. We can solve for the particular solution as follows:

$$y=e^{-2 \cos x-3}-3$$

Conclusion

In this article, we have explored the concept of a particular solution to a differential equation. We have focused on finding the particular solution to the differential equation $\frac{d y}{d x}=2 \sin (x)(3-y)$ with the initial condition $y(\pi)=2$. We have used the method of separation of variables and partial fractions to integrate both sides of the equation. We have applied the initial condition to find the particular solution, and we have solved for the particular solution. The particular solution is $y=e^{-2 \cos x-3}-3$.

Final Answer

The final answer is $y=e^{-2 \cos x-3}-3$.

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Introduction

In our previous article, we explored the concept of a particular solution to a differential equation. We focused on finding the particular solution to the differential equation $\frac{d y}{d x}=2 \sin (x)(3-y)$ with the initial condition $y(\pi)=2$. In this article, we will answer some frequently asked questions related to the particular solution.

Q1: What is the particular solution to the differential equation $\frac{d y}{d x}=2 \sin (x)(3-y)$?

A1: The particular solution to the differential equation $\frac{d y}{d x}=2 \sin (x)(3-y)$ is $y=e^{-2 \cos x-3}-3$.

Q2: How did you find the particular solution?

A2: We used the method of separation of variables and partial fractions to integrate both sides of the equation. We then applied the initial condition $y(\pi)=2$ to find the particular solution.

Q3: What is the initial condition?

A3: The initial condition is $y(\pi)=2$. This means that the value of $y$ at $x=\pi$ is 2.

Q4: How do you apply the initial condition?

A4: To apply the initial condition, we substitute the value of $y$ at $x=\pi$ into the general solution. In this case, we substituted $y(\pi)=2$ into the general solution to find the particular solution.

Q5: What is the general solution?

A5: The general solution is the solution to the differential equation without the initial condition. In this case, the general solution is $y=e^{-2 \cos x-3}-3$.

Q6: How do you find the general solution?

A6: We used the method of separation of variables and partial fractions to integrate both sides of the equation. We then expressed the left-hand side of the equation in terms of partial fractions and integrated both sides.

Q7: What is the difference between the particular solution and the general solution?

A7: The particular solution is the solution to the differential equation with the initial condition, while the general solution is the solution to the differential equation without the initial condition.

Q8: Can you explain the method of separation of variables?

A8: The method of separation of variables involves separating the unknown function and its derivative into different parts of the equation. In this case, we separated the variables as follows:

$$\frac{d y}{2 \sin (x)(3-y)}=d x$$

Q9: Can you explain partial fractions?

A9: Partial fractions involve expressing a fraction as a sum of simpler fractions. In this case, we expressed the left-hand side of the equation as follows:

$$\frac{1}{2 \sin (x)(3-y)}=\frac{A}{\sin (x)}+\frac{B}{3-y}$$

Q10: Can you provide more examples of differential equations?

A10: Yes, we can provide more examples of differential equations. However, the particular solution to the differential equation $\frac{d y}{d x}=2 \sin (x)(3-y)$ is a unique solution that satisfies the equation and the initial condition.

Conclusion

In this article, we have answered some frequently asked questions related to the particular solution to the differential equation $\frac{d y}{d x}=2 \sin (x)(3-y)$. We have explained the method of separation of variables, partial fractions, and the application of the initial condition. We have also provided examples of differential equations and explained the difference between the particular solution and the general solution.

Final Answer

The final answer is $y=e^{-2 \cos x-3}-3$.