Find The General Solution To E X D Y D X = 2 X 3 Y 3 E^x \frac{d Y}{d X} = 2 X^3 Y^3 E X D X D Y ​ = 2 X 3 Y 3 .

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Introduction

Separable differential equations are a type of differential equation that can be written in the form $\frac{d y}{d x} = f(x) g(y)$, where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$. These equations are called separable because we can separate the variables $x$ and $y$ and integrate both sides separately to find the general solution. In this article, we will find the general solution to the separable differential equation $e^x \frac{d y}{d x} = 2 x^3 y^3$.

Separating the Variables

To separate the variables, we need to isolate the terms involving $x$ on one side of the equation and the terms involving $y$ on the other side. We can do this by dividing both sides of the equation by $y^3$ and multiplying both sides by $e^{-x}$.

$$\frac{e^x \frac{d y}{d x}}{y^3} = 2 x^3$$

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

Integrating Both Sides

Now that we have separated the variables, we can integrate both sides of the equation. The left-hand side is a differential of the form $\frac{d y}{y^3}$, which can be integrated using the power rule of integration. The right-hand side is a product of two functions, $2 x^3$ and $e^{-x}$, which can be integrated using the product rule of integration.

$$\int \frac{d y}{y^3} = \int 2 x^3 e^{-x} d x$$

$$-\frac{1}{2 y^2} = -2 x^3 e^{-x} + C$$

Finding the General Solution

To find the general solution, we need to isolate $y$ on one side of the equation. We can do this by multiplying both sides of the equation by $-2$ and then taking the reciprocal of both sides.

$$\frac{1}{y^2} = 2 x^3 e^{-x} - C$$

$$y^2 = \frac{1}{2 x^3 e^{-x} - C}$$

$$y = \pm \sqrt{\frac{1}{2 x^3 e^{-x} - C}}$$

Conclusion

In this article, we found the general solution to the separable differential equation $e^x \frac{d y}{d x} = 2 x^3 y^3$. We separated the variables, integrated both sides, and isolated $y$ to find the general solution. The general solution is given by $y = \pm \sqrt{\frac{1}{2 x^3 e^{-x} - C}}$, where $C$ is an arbitrary constant.

Example

Let's consider an example to illustrate how to use the general solution. Suppose we want to find the solution to the initial value problem $e^x \frac{d y}{d x} = 2 x^3 y^3$ with the initial condition $y(0) = 1$. We can use the general solution to find the solution to this initial value problem.

$$y = \pm \sqrt{\frac{1}{2 x^3 e^{-x} - C}}$$

We are given the initial condition $y(0) = 1$, so we can substitute $x = 0$ and $y = 1$ into the general solution to find the value of $C$.

$$1 = \pm \sqrt{\frac{1}{2 (0)^3 e^{-0} - C}}$$

$$1 = \pm \sqrt{\frac{1}{-C}}$$

$$1 = \pm \frac{1}{\sqrt{-C}}$$

$$\sqrt{-C} = 1$$

$$-C = 1$$

$$C = -1$$

Now that we have found the value of $C$, we can substitute it into the general solution to find the solution to the initial value problem.

$$y = \pm \sqrt{\frac{1}{2 x^3 e^{-x} - (-1)}}$$

$$y = \pm \sqrt{\frac{1}{2 x^3 e^{-x} + 1}}$$

Graphing the Solution

To visualize the solution, we can graph the function $y = \pm \sqrt{\frac{1}{2 x^3 e^{-x} + 1}}$. We can use a graphing calculator or software to graph the function.

The graph of the function $y = \pm \sqrt{\frac{1}{2 x^3 e^{-x} + 1}}$ is a pair of curves that intersect at the point $(0, 1)$. The curves are symmetric about the $x$-axis and have a minimum value of $0$ at $x = 0$.

Conclusion

In this article, we found the general solution to the separable differential equation $e^x \frac{d y}{d x} = 2 x^3 y^3$. We separated the variables, integrated both sides, and isolated $y$ to find the general solution. We also considered an example to illustrate how to use the general solution to find the solution to an initial value problem. Finally, we graphed the solution to visualize the behavior of the function.

Applications

Separable differential equations have many applications in physics, engineering, and other fields. Some examples of applications include:

• Population growth: Separable differential equations can be used to model population growth and decline.
• Chemical reactions: Separable differential equations can be used to model chemical reactions and the rates of reaction.
• Electric circuits: Separable differential equations can be used to model electric circuits and the behavior of electrical currents.
• Optics: Separable differential equations can be used to model optical systems and the behavior of light.

Conclusion

In conclusion, separable differential equations are a powerful tool for modeling and solving problems in physics, engineering, and other fields. By separating the variables, integrating both sides, and isolating $y$, we can find the general solution to a separable differential equation. We also considered an example to illustrate how to use the general solution to find the solution to an initial value problem. Finally, we graphed the solution to visualize the behavior of the function.

Introduction

Separable differential equations are a type of differential equation that can be written in the form $\frac{d y}{d x} = f(x) g(y)$, where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$. These equations are called separable because we can separate the variables $x$ and $y$ and integrate both sides separately to find the general solution. In this article, we will answer some common questions about 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 $\frac{d y}{d x} = f(x) g(y)$, 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, we need to check if it can be written in the form $\frac{d y}{d x} = f(x) g(y)$. If it can be written in this form, then it is a separable differential equation.

Q: How do I separate the variables in a separable differential equation?

A: To separate the variables in a separable differential equation, we need to isolate the terms involving $x$ on one side of the equation and the terms involving $y$ on the other side. We can do this by dividing both sides of the equation by $y^3$ and multiplying both sides by $e^{-x}$.

Q: How do I integrate both sides of a separable differential equation?

A: To integrate both sides of a separable differential equation, we need to integrate the left-hand side, which is a differential of the form $\frac{d y}{y^3}$, and the right-hand side, which is a product of two functions, $2 x^3$ and $e^{-x}$.

Q: What is the general solution to a separable differential equation?

A: The general solution to a separable differential equation is given by $y = \pm \sqrt{\frac{1}{2 x^3 e^{-x} - C}}$, where $C$ is an arbitrary constant.

Q: How do I find the value of the arbitrary constant $C$?

A: To find the value of the arbitrary constant $C$, we need to use the initial condition of the differential equation. We can substitute the initial condition into the general solution and solve for $C$.

Q: Can I use a graphing calculator or software to graph the solution to a separable differential equation?

A: Yes, you can use a graphing calculator or software to graph the solution to a separable differential equation. This can help you visualize the behavior of the function and understand the solution better.

Q: What are some applications of separable differential equations?

A: Separable differential equations have many applications in physics, engineering, and other fields. Some examples of applications include population growth, chemical reactions, electric circuits, and optics.

Q: Can I use separable differential equations to model real-world problems?

A: Yes, you can use separable differential equations to model real-world problems. By separating the variables, integrating both sides, and isolating $y$, we can find the general solution to a separable differential equation and use it to model real-world problems.

Q: Are there any limitations to using separable differential equations?

A: Yes, there are some limitations to using separable differential equations. For example, not all differential equations can be written in the form $\frac{d y}{d x} = f(x) g(y)$, and some separable differential equations may not have a general solution that can be expressed in terms of elementary functions.

Conclusion

In conclusion, separable differential equations are a powerful tool for modeling and solving problems in physics, engineering, and other fields. By separating the variables, integrating both sides, and isolating $y$, we can find the general solution to a separable differential equation. We also answered some common questions about separable differential equations and discussed some applications and limitations of using these equations.

Frequently Asked Questions

• Q: What is the difference between a separable differential equation and a non-separable differential equation? A: A separable differential equation is a type of differential equation that can be written in the form $\frac{d y}{d x} = f(x) g(y)$, where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$. A non-separable differential equation is a type of differential equation that cannot be written in this form.
• Q: Can I use a computer algebra system to solve a separable differential equation? A: Yes, you can use a computer algebra system to solve a separable differential equation. These systems can help you separate the variables, integrate both sides, and isolate $y$ to find the general solution.
• Q: Are there any other types of differential equations that are similar to separable differential equations? A: Yes, there are other types of differential equations that are similar to separable differential equations. For example, linear differential equations and Bernoulli differential equations are also types of differential equations that can be solved using similar techniques.

Glossary

• Separable differential equation: A type of differential equation that can be written in the form $\frac{d y}{d x} = f(x) g(y)$, where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$.
• General solution: The solution to a differential equation that is expressed in terms of an arbitrary constant.
• Arbitrary constant: A constant that is used to represent an unknown value in a differential equation.
• Initial condition: A condition that is used to determine the value of an arbitrary constant in a differential equation.
• Graphing calculator: A calculator that is used to graph functions and visualize the behavior of a differential equation.
• Computer algebra system: A system that is used to solve mathematical problems, including differential equations.