A. Find The Differential Equation Of The Family Of Curves Y = A E 2 X + B E − 3 X Y = A E^{2x} + B E^{-3x} Y = A E 2 X + B E − 3 X For Different Values Of A A A And B B B .Answer: D 2 Y D X 2 + D Y D X − 6 Y = 0 \frac{d^2 Y}{dx^2} + \frac{dy}{dx} - 6y = 0 D X 2 D 2 Y + D X D Y − 6 Y = 0

Mar 9, 2025 by ADMIN 292 views

**22. a. Finding the Differential Equation of a Family of Curves**

Introduction

In this article, we will explore the concept of finding the differential equation of a family of curves. A family of curves is a set of curves that are related to each other through a common equation. The differential equation of a family of curves is a mathematical equation that describes the relationship between the variables of the curve and their derivatives. In this case, we will find the differential equation of the family of curves given by the equation $y = A e^{2x} + B e^{-3x}$ for different values of $A$ and $B$ .

The Family of Curves

The family of curves is given by the equation $y = A e^{2x} + B e^{-3x}$ , where $A$ and $B$ are constants. This equation represents a set of curves that are related to each other through the common exponential functions $e^{2x}$ and $e^{-3x}$ . The values of $A$ and $B$ determine the shape and position of the curves.

Finding the Differential Equation

To find the differential equation of the family of curves, we need to find the first and second derivatives of the equation with respect to $x$ . We will then use these derivatives to form the differential equation.

First Derivative

The first derivative of the equation $y = A e^{2x} + B e^{-3x}$ with respect to $x$ is given by:

\frac{dy}{dx} = 2A e^{2x} - 3B e^{-3x}

Second Derivative

The second derivative of the equation $y = A e^{2x} + B e^{-3x}$ with respect to $x$ is given by:

\frac{d^2 y}{dx^2} = 4A e^{2x} + 9B e^{-3x}

Differential Equation

Now that we have found the first and second derivatives, we can form the differential equation. We will use the following equation:

\frac{d^2 y}{dx^2} + \frac{dy}{dx} - 6y = 0

This equation is a second-order linear homogeneous differential equation. It describes the relationship between the variables of the curve and their derivatives.

Discussion

The differential equation $\frac{d^2 y}{dx^2} + \frac{dy}{dx} - 6y = 0$ is a fundamental equation in mathematics. It is used to describe the behavior of a wide range of physical systems, including electrical circuits, mechanical systems, and population dynamics.

The equation is a second-order linear homogeneous differential equation, which means that it has a general solution of the form:

y = c_1 e^{r_1 x} + c_2 e^{r_2 x}

where $c_1$ and $c_2$ are constants, and $r_1$ and $r_2$ are the roots of the characteristic equation.

The characteristic equation of the differential equation is given by:

r^2 + r - 6 = 0

Solving this equation, we get:

r = 2, -3

Therefore, the general solution of the differential equation is:

y = c_1 e^{2x} + c_2 e^{-3x}

This solution represents the family of curves given by the equation $y = A e^{2x} + B e^{-3x}$ for different values of $A$ and $B$ .

Conclusion

In this article, we have found the differential equation of the family of curves given by the equation $y = A e^{2x} + B e^{-3x}$ for different values of $A$ and $B$ . The differential equation is a second-order linear homogeneous differential equation, which describes the relationship between the variables of the curve and their derivatives. The general solution of the differential equation represents the family of curves, and it is a fundamental equation in mathematics.

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.
22. a. Finding the Differential Equation of a Family of Curves: Q&A ====================================================================

Introduction

In our previous article, we explored the concept of finding the differential equation of a family of curves. We found the differential equation of the family of curves given by the equation $y = A e^{2x} + B e^{-3x}$ for different values of $A$ and $B$ . In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the significance of finding the differential equation of a family of curves?

A: Finding the differential equation of a family of curves is significant because it helps us understand the behavior of the curves and their derivatives. It also helps us to identify the underlying mathematical structure of the curves.

Q: How do we find the differential equation of a family of curves?

A: To find the differential equation of a family of curves, we need to find the first and second derivatives of the equation with respect to $x$ . We then use these derivatives to form the differential equation.

Q: What is the characteristic equation of the differential equation?

A: The characteristic equation of the differential equation is given by $r^2 + r - 6 = 0$ . Solving this equation, we get $r = 2, -3$ .

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

A: The general solution of the differential equation is given by $y = c_1 e^{2x} + c_2 e^{-3x}$ , where $c_1$ and $c_2$ are constants.

Q: How do we use the differential equation to describe the behavior of the curves?

A: We can use the differential equation to describe the behavior of the curves by analyzing the roots of the characteristic equation. If the roots are real and distinct, the curves will be oscillatory. If the roots are complex conjugates, the curves will be exponential.

Q: Can we use the differential equation to solve other types of problems?

A: Yes, we can use the differential equation to solve other types of problems, such as electrical circuits, mechanical systems, and population dynamics.

Q: How do we apply the differential equation to real-world problems?

A: We can apply the differential equation to real-world problems by identifying the underlying mathematical structure of the problem and using the differential equation to model the behavior of the system.

Q: What are some common applications of the differential equation?

A: Some common applications of the differential equation include:

Electrical circuits: The differential equation can be used to model the behavior of electrical circuits, such as RC circuits and LC circuits.
Mechanical systems: The differential equation can be used to model the behavior of mechanical systems, such as springs and masses.
Population dynamics: The differential equation can be used to model the behavior of populations, such as the growth and decline of populations.

Conclusion

In this article, we have answered some frequently asked questions related to finding the differential equation of a family of curves. We have discussed the significance of finding the differential equation, how to find it, and how to use it to describe the behavior of the curves. We have also discussed some common applications of the differential equation and how to apply it to real-world problems.

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.