Find The General Solution Of The Differential Equation:$\[ Y^{\prime}(t) = 4 + E^{-3t} \\]$\[ Y(t) = \square \\]

Feb 25, 2025 by ADMIN 113 views

**Finding the General Solution of a Differential Equation**

Introduction

Differential equations are a fundamental concept in mathematics, and they play a crucial role in modeling various phenomena in physics, engineering, and other fields. In this article, we will focus on finding the general solution of a given differential equation, which is a key aspect of solving differential equations. The differential equation we will be dealing with is $y^{\prime}(t) = 4 + e^{-3t}$ , where $y(t)$ is the unknown function we need to find.

Understanding the Differential Equation

The given differential equation is a first-order differential equation, which means it involves a first derivative of the unknown function $y(t)$ . The equation is $y^{\prime}(t) = 4 + e^{-3t}$ , where $y^{\prime}(t)$ represents the derivative of $y(t)$ with respect to $t$ . The right-hand side of the equation is a sum of two terms: a constant term $4$ and an exponential term $e^{-3t}$ .

Separating the Variables

To find the general solution of the differential equation, we need to separate the variables. This involves rearranging the equation so that all the terms involving $y(t)$ are on one side of the equation, and all the terms involving $t$ are on the other side. We can do this by subtracting $4$ from both sides of the equation and then multiplying both sides by $e^{3t}$ .

${ y^{\prime}(t) - 4 = e^{-3t} }$

Multiplying both sides by $e^{3t}$ :

${ e^{3t}y^{\prime}(t) - 4e^{3t} = 1 }$

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 derivative of $y(t)$ , which we can integrate with respect to $t$ . The right-hand side of the equation is a constant term, which we can integrate as well.

${ \int e^{3t}y^{\prime}(t) dt - \int 4e^{3t} dt = \int 1 dt }$

Evaluating the Integrals

Evaluating the integrals on both sides of the equation, we get:

${ e^{3t}y(t) - \int 4e^{3t} dt = t + C }$

where $C$ is the constant of integration.

Simplifying the Equation

Now that we have evaluated the integrals, we can simplify the equation by combining like terms. We can do this by adding $\int 4e^{3t} dt$ to both sides of the equation.

${ e^{3t}y(t) = t + C + \int 4e^{3t} dt }$

Finding the General Solution

To find the general solution of the differential equation, we need to isolate $y(t)$ on one side of the equation. We can do this by dividing both sides of the equation by $e^{3t}$ .

${ y(t) = \frac{t + C + \int 4e^{3t} dt}{e^{3t}} }$

Evaluating the Integral

To evaluate the integral $\int 4e^{3t} dt$ , we can use the formula for the integral of an exponential function.

${ \int 4e^{3t} dt = \frac{4}{3}e^{3t} + C_1 }$

where $C_1$ is another constant of integration.

Substituting the Integral

Substituting the integral into the equation for $y(t)$ , we get:

${ y(t) = \frac{t + C + \frac{4}{3}e^{3t} + C_1}{e^{3t}} }$

Simplifying the Equation

Now that we have substituted the integral, we can simplify the equation by combining like terms.

${ y(t) = \frac{t + C + \frac{4}{3}e^{3t} + C_1}{e^{3t}} }$

Finding the General Solution

To find the general solution of the differential equation, we need to isolate $y(t)$ on one side of the equation. We can do this by dividing both sides of the equation by $e^{3t}$ .

${ y(t) = t + C + \frac{4}{3} + \frac{C_1}{e^{3t}} }$

Conclusion

In this article, we have found the general solution of the differential equation $y^{\prime}(t) = 4 + e^{-3t}$ . The general solution is given by the equation $y(t) = t + C + \frac{4}{3} + \frac{C_1}{e^{3t}}$ , where $C$ and $C_1$ are constants of integration. This solution represents all possible solutions of the differential equation, and it can be used to model various phenomena in physics, engineering, and other fields.

Final Answer

The final answer is: $\boxed{y(t) = t + C + \frac{4}{3} + \frac{C_1}{e^{3t}}}$

Introduction

In our previous article, we discussed how to find the general solution of a differential equation. In this article, we will answer some common questions related to finding the general solution of a differential equation.

Q: What is the general solution of a differential equation?

A: The general solution of a differential equation is a solution that contains all possible solutions of the equation. It is a function that satisfies the differential equation for all values of the independent variable.

Q: How do I find the general solution of a differential equation?

A: To find the general solution of a differential equation, you need to follow these steps:

Separate the variables by rearranging the equation so that all the terms involving the unknown function are on one side of the equation, and all the terms involving the independent variable are on the other side.
Integrate both sides of the equation to eliminate the derivative.
Evaluate the integrals and simplify the equation.
Isolate the unknown function on one side of the equation.

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

A: The general solution of a differential equation is a solution that contains all possible solutions of the equation, while the particular solution is a specific solution that satisfies the initial conditions of the problem.

Q: How do I find the particular solution of a differential equation?

A: To find the particular solution of a differential equation, you need to follow these steps:

Find the general solution of the differential equation.
Apply the initial conditions of the problem to the general solution.
Solve for the constants of integration.

Q: What is the role of the constants of integration in finding the general solution of a differential equation?

A: The constants of integration are used to represent the arbitrary constants that arise when integrating the differential equation. They are used to satisfy the initial conditions of the problem and to ensure that the solution satisfies the differential equation.

Q: How do I determine the number of constants of integration in a differential equation?

A: The number of constants of integration in a differential equation is equal to the order of the differential equation. For example, if the differential equation is of order 2, then there will be 2 constants of integration.

Q: What is the significance of the initial conditions in finding the particular solution of a differential equation?

A: The initial conditions are used to determine the particular solution of a differential equation. They provide the necessary information to solve for the constants of integration and to find the specific solution that satisfies the problem.

Q: How do I apply the initial conditions to the general solution of a differential equation?

A: To apply the initial conditions to the general solution of a differential equation, you need to substitute the initial conditions into the general solution and solve for the constants of integration.

Q: What is the final answer to the differential equation?

A: The final answer to the differential equation is the particular solution that satisfies the initial conditions of the problem. It is a specific solution that represents the actual behavior of the system.

Conclusion

In this article, we have answered some common questions related to finding the general solution of a differential equation. We have discussed the role of the constants of integration, the significance of the initial conditions, and the steps involved in finding the particular solution of a differential equation. We hope that this article has provided a clear understanding of the concepts involved in finding the general solution of a differential equation.

Final Answer

The final answer is: $\boxed{y(t) = t + C + \frac{4}{3} + \frac{C_1}{e^{3t}}}$