Solve The Differential Equation:${ \frac{d Y}{d X} + 3 Y = E^{4 X} }$

Mar 9, 2025 by ADMIN 71 views

**Solving Differential Equations: A Step-by-Step Guide**

Introduction

Differential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of differential equation, which is a first-order linear differential equation. We will use the equation $\frac{d y}{d x} + 3 y = e^{4 x}$ as an example and provide a step-by-step guide on how to solve it.

What is a Differential Equation?

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. Differential equations are used to model a wide range of phenomena, including population growth, chemical reactions, electrical circuits, and more.

Types of Differential Equations

There are several types of differential equations, including:

First-order differential equations: These equations involve a first derivative of the unknown function.
Second-order differential equations: These equations involve a second derivative of the unknown function.
Linear differential equations: These equations involve a linear combination of the unknown function and its derivatives.
Nonlinear differential equations: These equations involve a nonlinear combination of the unknown function and its derivatives.

The Equation We Will Solve

The equation we will solve is a first-order linear differential equation:

$\frac{d y}{d x} + 3 y = e^{4 x}$

This equation involves a first derivative of the unknown function $y$ and a linear combination of $y$ and its derivative.

Step 1: Identify the Type of Differential Equation

The equation we are given is a first-order linear differential equation. This means that we can use the method of integrating factors to solve it.

Step 2: Find the Integrating Factor

The integrating factor for a first-order linear differential equation is given by:

$I(x) = e^{\int P(x) dx}$

where $P(x)$ is the coefficient of the derivative term. In this case, $P(x) = 3$ , so the integrating factor is:

$I(x) = e^{\int 3 dx} = e^{3x}$

Step 3: Multiply Both Sides by the Integrating Factor

We multiply both sides of the equation by the integrating factor:

$e^{3x} \frac{d y}{d x} + 3 e^{3x} y = e^{4x}$

Step 4: Recognize the Product Rule

The left-hand side of the equation is a product of two functions, $e^{3x}$ and $y$ . We can recognize the product rule for differentiation:

$\frac{d}{dx} (e^{3x} y) = e^{3x} \frac{d y}{d x} + 3 e^{3x} y$

Step 5: Integrate Both Sides

We can now integrate both sides of the equation:

$\int \frac{d}{dx} (e^{3x} y) dx = \int e^{4x} dx$

Using the product rule, we can rewrite the left-hand side as:

$\int e^{3x} \frac{d y}{d x} dx + \int 3 e^{3x} y dx = \int e^{4x} dx$

Step 6: Evaluate the Integrals

We can now evaluate the integrals:

$e^{3x} y + \int 3 e^{3x} y dx = \frac{1}{4} e^{4x} + C$

where $C$ is the constant of integration.

Step 7: Solve for y

We can now solve for $y$ :

$e^{3x} y = \frac{1}{4} e^{4x} + C$

Dividing both sides by $e^{3x}$ , we get:

$y = \frac{1}{4} e^{x} + C e^{-3x}$

Conclusion

In this article, we solved a first-order linear differential equation using the method of integrating factors. We identified the type of differential equation, found the integrating factor, multiplied both sides by the integrating factor, recognized the product rule, integrated both sides, evaluated the integrals, and solved for $y$ . This is a general method for solving first-order linear differential equations, and it can be applied to a wide range of problems.

Applications of Differential Equations

Differential equations have a wide range of applications in science, engineering, economics, and other fields. Some examples include:

Population growth: Differential equations can be used to model population growth and decline.
Chemical reactions: Differential equations can be used to model chemical reactions and predict the concentration of reactants and products.
Electrical circuits: Differential equations can be used to model electrical circuits and predict the behavior of electrical systems.
Economics: Differential equations can be used to model economic systems and predict the behavior of economic variables such as GDP and inflation.

Final Thoughts

Introduction

In our previous article, we solved a first-order linear differential equation using the method of integrating factors. However, we know that differential equations can be complex and challenging to solve. In this article, we will provide a Q&A guide to help you better understand differential equations and how to solve them.

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 are the different types of differential equations?

A: There are several types of differential equations, including:

First-order differential equations: These equations involve a first derivative of the unknown function.
Second-order differential equations: These equations involve a second derivative of the unknown function.
Linear differential equations: These equations involve a linear combination of the unknown function and its derivatives.
Nonlinear differential equations: These equations involve a nonlinear combination of the unknown function and its derivatives.

Q: How do I know which method to use to solve a differential equation?

A: The method you use to solve a differential equation depends on the type of equation and its complexity. Some common methods include:

Separation of variables: This method is used to solve first-order differential equations.
Integrating factors: This method is used to solve first-order linear differential equations.
Undetermined coefficients: This method is used to solve second-order linear differential equations.
Variation of parameters: This method is used to solve second-order linear differential equations.

Q: What is the integrating factor method?

A: The integrating factor method is a technique used to solve first-order linear differential equations. It involves finding an integrating factor, which is a function that, when multiplied by the differential equation, makes it exact.

Q: How do I find the integrating factor?

A: To find the integrating factor, you need to integrate the coefficient of the derivative term. The integrating factor is given by:

$I(x) = e^{\int P(x) dx}$

where $P(x)$ is the coefficient of the derivative term.

Q: What is the product rule?

A: The product rule is a mathematical rule that states that the derivative of a product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Q: How do I use the product rule to solve a differential equation?

A: To use the product rule to solve a differential equation, you need to recognize the product rule in the differential equation. Once you have recognized the product rule, you can integrate both sides of the equation to solve for the unknown function.

Q: What are some common applications of differential equations?

A: Differential equations have a wide range of applications in science, engineering, economics, and other fields. Some examples include:

Population growth: Differential equations can be used to model population growth and decline.
Chemical reactions: Differential equations can be used to model chemical reactions and predict the concentration of reactants and products.
Electrical circuits: Differential equations can be used to model electrical circuits and predict the behavior of electrical systems.
Economics: Differential equations can be used to model economic systems and predict the behavior of economic variables such as GDP and inflation.

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

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

Not recognizing the type of differential equation: Make sure you recognize the type of differential equation you are dealing with, as this will determine the method you use to solve it.
Not using the correct method: Make sure you use the correct method to solve the differential equation.
Not checking your work: Make sure you check your work to ensure that you have solved the differential equation correctly.

Conclusion

In this article, we provided a Q&A guide to help you better understand differential equations and how to solve them. We covered topics such as the different types of differential equations, the integrating factor method, the product rule, and common applications of differential equations. We also provided some common mistakes to avoid when solving differential equations. By following this guide, you should be able to better understand and solve differential equations.