Substituting the Value of C

Oct 13, 2025 by windailys.com 58 views

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Introduction

Differential equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving a specific differential equation, which is given by:

\frac{d y}{d x} = x^4(y - 2)

Given that $f(0) = 0$ , we will use various techniques to find the solution to this differential equation.

Separation of Variables

One of the most common methods for solving differential equations is the separation of variables. This method involves separating the variables $x$ and $y$ on either side of the equation.

To separate the variables, we can rewrite the given differential equation as:

\frac{d y}{y - 2} = x^4 dx

Now, we can integrate both sides of the equation to find the solution.

Integration

To integrate the left-hand side of the equation, we can use the following formula:

\int \frac{d y}{y - 2} = \ln|y - 2| + C

where $C$ is the constant of integration.

To integrate the right-hand side of the equation, we can use the following formula:

\int x^4 dx = \frac{x^5}{5} + C

Now, we can equate the two expressions and solve for $y$ .

Solving for y

Equating the two expressions, we get:

\ln|y - 2| = \frac{x^5}{5} + C

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

|y - 2| = e^{\frac{x^5}{5} + C}

Since $f(0) = 0$ , we know that $y(0) = 0$ . Therefore, we can substitute $x = 0$ and $y = 0$ into the equation to find the value of $C$ .

Finding the Value of C

Substituting $x = 0$ and $y = 0$ into the equation, we get:

|0 - 2| = e^{\frac{0^5}{5} + C}

Simplifying the equation, we get:

2 = e^C

Taking the natural logarithm of both sides, we get:

\ln 2 = C

Substituting the Value of C

Substituting the value of $C$ into the equation, we get:

|y - 2| = e^{\frac{x^5}{5} + \ln 2}

Simplifying the equation, we get:

|y - 2| = 2e^{\frac{x^5}{5}}

Solving for y

To solve for $y$ , we can remove the absolute value sign by considering two cases:

Case 1: $y - 2 \geq 0$

In this case, we can remove the absolute value sign and solve for $y$ :

y - 2 = 2e^{\frac{x^5}{5}}

Solving for $y$ , we get:

y = 2 + 2e^{\frac{x^5}{5}}

Case 2: $y - 2 < 0$

In this case, we can remove the absolute value sign and solve for $y$ :

-(y - 2) = 2e^{\frac{x^5}{5}}

Simplifying the equation, we get:

-y + 2 = 2e^{\frac{x^5}{5}}

Solving for $y$ , we get:

y = 2 - 2e^{\frac{x^5}{5}}

Conclusion

In this article, we have solved the differential equation $\frac{d y}{d x} = x^4(y - 2)$ given that $f(0) = 0$ . We used the method of separation of variables to separate the variables $x$ and $y$ on either side of the equation. We then integrated both sides of the equation to find the solution. Finally, we solved for $y$ by considering two cases: $y - 2 \geq 0$ and $y - 2 < 0$ .

The final solution to the differential equation is:

y = 2 + 2e^{\frac{x^5}{5}} \text{ or } y = 2 - 2e^{\frac{x^5}{5}}

This solution satisfies the initial condition $f(0) = 0$ .

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.

Glossary

Differential equation: An equation that involves an unknown function and its derivatives.
Separation of variables: A method for solving differential equations by separating the variables on either side of the equation.
Integration: The process of finding the antiderivative of a function.
Constant of integration: A constant that is added to the solution of a differential equation to make it unique.
Exponentiation: The process of raising a number to a power.
Absolute value: The distance of a number from zero on the number line.

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Q: What is a differential equation?

A: A differential equation is an equation that involves an unknown function and its derivatives. It is a mathematical equation that describes how a quantity changes over time or space.

Q: What is the method of separation of variables?

A: The method of separation of variables is a technique used to solve differential equations by separating the variables on either side of the equation. This involves 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 know if a differential equation can be solved using the method of separation of variables?

A: A differential equation can be solved using the method of separation of variables if it can be written in the form:

\frac{dy}{dx} = f(x)g(y)

where $f(x)$ and $g(y)$ are functions of $x$ and $y$ respectively.

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

A: A first-order differential equation is a differential equation that involves the first derivative of the unknown function. A higher-order differential equation is a differential equation that involves the second or higher derivative of the unknown function.

Q: How do I solve a differential equation that involves a second or higher derivative?

A: To solve a differential equation that involves a second or higher derivative, you can use various techniques such as:

Substitution: Substitute the unknown function and its derivatives into a new equation.
Integration: Integrate the equation to find the antiderivative of the unknown function.
Differentiation: Differentiate the equation to find the derivative of the unknown function.

Q: What is the significance of the initial condition in solving a differential equation?

A: The initial condition is a condition that is specified at a particular point in time or space. It is used to determine the value of the unknown function at that point. The initial condition is essential in solving a differential equation because it provides a starting point for the solution.

Q: Can a differential equation have multiple solutions?

A: Yes, a differential equation can have multiple solutions. This occurs when the differential equation has multiple initial conditions or when the solution is not unique.

Q: How do I determine if a differential equation has a unique solution?

A: To determine if a differential equation has a unique solution, you can use various techniques such as:

Analyzing the equation: Analyze the equation to determine if it has a unique solution.
Using numerical methods: Use numerical methods such as the Euler method or the Runge-Kutta method to approximate the solution.
Using graphical methods: Use graphical methods such as plotting the solution to determine if it is unique.

Q: What is the importance of differential equations in real-world applications?

A: Differential equations are essential in various real-world applications such as:

Physics: Differential equations are used to describe the motion of objects, the behavior of electrical circuits, and the properties of materials.
Engineering: Differential equations are used to design and optimize systems, to model the behavior of complex systems, and to predict the performance of systems.
Biology: Differential equations are used to model the growth and behavior of populations, to study the spread of diseases, and to understand the behavior of complex biological systems.

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

A: Yes, you can use a computer to solve a differential equation. There are various software packages and programming languages available that can be used to solve differential equations, such as:

MATLAB: A high-level programming language that is widely used for numerical computation and data analysis.
Python: A high-level programming language that is widely used for numerical computation and data analysis.
Mathematica: A computer algebra system that is widely used for symbolic and numerical computation.

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

A: Some common mistakes to avoid when solving a differential equation include:

Not checking the initial condition: Failing to check the initial condition can lead to incorrect solutions.
Not using the correct method: Using the wrong method can lead to incorrect solutions.
Not checking the solution: Failing to check the solution can lead to incorrect conclusions.

Q: How do I know if I have solved a differential equation correctly?

A: To determine if you have solved a differential equation correctly, you can use various techniques such as:

Checking the initial condition: Check that the solution satisfies the initial condition.
Checking the solution: Check that the solution satisfies the differential equation.
Using numerical methods: Use numerical methods to approximate the solution and compare it with the analytical solution.

Q: Can I use a differential equation to model a real-world problem?

A: Yes, you can use a differential equation to model a real-world problem. Differential equations are widely used in various fields such as physics, engineering, and biology to model complex systems and phenomena.

Q: How do I choose the right differential equation to model a real-world problem?

A: To choose the right differential equation to model a real-world problem, you can use various techniques such as:

Analyzing the problem: Analyze the problem to determine the type of differential equation that is required.
Using mathematical models: Use mathematical models to determine the type of differential equation that is required.
Consulting with experts: Consult with experts in the field to determine the type of differential equation that is required.

Q: Can I use a differential equation to make predictions about a real-world problem?

A: Yes, you can use a differential equation to make predictions about a real-world problem. Differential equations are widely used in various fields such as physics, engineering, and biology to make predictions about complex systems and phenomena.

Q: How do I use a differential equation to make predictions about a real-world problem?

A: To use a differential equation to make predictions about a real-world problem, you can use various techniques such as:

Analyzing the problem: Analyze the problem to determine the type of differential equation that is required.
Using mathematical models: Use mathematical models to determine the type of differential equation that is required.
Consulting with experts: Consult with experts in the field to determine the type of differential equation that is required.

Q: Can I use a differential equation to optimize a real-world problem?

A: Yes, you can use a differential equation to optimize a real-world problem. Differential equations are widely used in various fields such as physics, engineering, and biology to optimize complex systems and phenomena.

Q: How do I use a differential equation to optimize a real-world problem?

A: To use a differential equation to optimize a real-world problem, you can use various techniques such as:

Analyzing the problem: Analyze the problem to determine the type of differential equation that is required.
Using mathematical models: Use mathematical models to determine the type of differential equation that is required.
Consulting with experts: Consult with experts in the field to determine the type of differential equation that is required.

Q: Can I use a differential equation to understand a real-world problem?

A: Yes, you can use a differential equation to understand a real-world problem. Differential equations are widely used in various fields such as physics, engineering, and biology to understand complex systems and phenomena.

Q: How do I use a differential equation to understand a real-world problem?

A: To use a differential equation to understand a real-world problem, you can use various techniques such as:

Analyzing the problem: Analyze the problem to determine the type of differential equation that is required.
Using mathematical models: Use mathematical models to determine the type of differential equation that is required.
Consulting with experts: Consult with experts in the field to determine the type of differential equation that is required.