Find The Particular Solution Of The Differential Equation That Satisfies The Initial Condition. Enter Your Solution As An Equation.

Mar 13, 2025 by ADMIN 132 views

**Finding Particular Solutions to Differential Equations: A Step-by-Step Guide**

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Introduction

Differential equations are a fundamental concept in calculus, and solving them is crucial in various fields such as physics, engineering, and economics. In this article, we will focus on finding the particular solution of a given differential equation that satisfies a specific initial condition. We will use the differential equation $x + yy' = 0$ and the initial condition $y(25) = 25$ as a case study.

Understanding Differential Equations

A differential equation is an equation that involves an unknown function and its derivatives. It is a mathematical statement that describes how a quantity changes over time or space. In this case, we have the differential equation $x + yy' = 0$ , where $y$ is the unknown function and $y'$ is its derivative.

Separation of Variables

To solve the differential equation, we can use the method of separation of variables. This method involves separating the variables $x$ and $y$ into different parts of the equation. We can rewrite the equation as:

\frac{dy}{dx} = -\frac{x}{y}

Integrating Both Sides

Next, we can integrate both sides of the equation with respect to $x$ . This will give us:

\int \frac{dy}{dx} dx = \int -\frac{x}{y} dx

Evaluating the Integrals

Evaluating the integrals, we get:

y = \int -\frac{x}{y} dx

Using the Initial Condition

We are given the initial condition $y(25) = 25$ . We can use this condition to find the particular solution of the differential equation. Substituting $x = 25$ and $y = 25$ into the equation, we get:

25 = \int -\frac{25}{25} dx

Simplifying the Equation

Simplifying the equation, we get:

25 = \int -1 dx

Evaluating the Integral

Evaluating the integral, we get:

25 = -x + C

Using the Initial Condition Again

We can use the initial condition again to find the value of $C$ . Substituting $x = 25$ and $y = 25$ into the equation, we get:

25 = -25 + C

Solving for C

Solving for $C$ , we get:

C = 50

Substituting C Back into the Equation

Substituting $C = 50$ back into the equation, we get:

25 = -x + 50

Simplifying the Equation

Simplifying the equation, we get:

x = 25

Finding the Particular Solution

We can now find the particular solution of the differential equation. Substituting $x = 25$ into the equation, we get:

y = 25

Conclusion

In this article, we have found the particular solution of the differential equation $x + yy' = 0$ that satisfies the initial condition $y(25) = 25$ . The particular solution is $y = 25$ . This solution satisfies the initial condition and is a valid solution to the differential equation.

Final Answer

The final answer is $\boxed{y = 25}$ .

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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 statement 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. It involves separating the variables into different parts of the equation, allowing us to integrate both sides and find the solution.

Q: How do I use the initial condition to find the particular solution?

A: To use the initial condition to find the particular solution, substitute the given values of $x$ and $y$ into the equation and solve for the unknown constant $C$ . Then, substitute the value of $C$ back into the equation to find the particular solution.

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

A: A particular solution is a specific solution to a differential equation that satisfies a given initial condition. A general solution, on the other hand, is a family of solutions that satisfy the differential equation, but may not satisfy a specific initial condition.

Q: How do I know if I have found the correct particular solution?

A: To verify that you have found the correct particular solution, substitute the values of $x$ and $y$ back into the differential equation and check if the equation is satisfied. If it is, then you have found the correct particular solution.

Q: Can I use the method of separation of variables to solve any differential equation?

A: No, the method of separation of variables can only be used to solve differential equations that can be written in the form $\frac{dy}{dx} = f(x,y)$ . If the differential equation cannot be written in this form, then the method of separation of variables cannot be used.

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

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

Not checking if the initial condition is satisfied
Not verifying that the solution satisfies the differential equation
Not using the correct method to solve the differential equation
Not checking for any singularities or discontinuities in the solution

Q: How do I know if I have found the correct general solution?

A: To verify that you have found the correct general solution, substitute the solution back into the differential equation and check if the equation is satisfied. If it is, then you have found the correct general solution.

Q: Can I use a calculator or computer to solve differential equations?

A: Yes, you can use a calculator or computer to solve differential equations. Many calculators and computer software programs, such as Mathematica or MATLAB, have built-in functions for solving differential equations.

Q: What are some real-world applications of differential equations?

A: Differential equations have many real-world applications, including:

Modeling population growth and decline
Describing the motion of objects under the influence of gravity or other forces
Modeling the spread of diseases
Describing the behavior of electrical circuits
Modeling the behavior of financial markets

Conclusion

In this article, we have answered some frequently asked questions about finding particular solutions to differential equations. We have discussed the method of separation of variables, how to use the initial condition to find the particular solution, and some common mistakes to avoid when solving differential equations. We have also discussed some real-world applications of differential equations.