Which Of The Following Is A Solution To The Differential Equation 1 Y ′ ′ − 4 Y = 0 1 Y^{\prime \prime}-4 Y=0 1 Y ′′ − 4 Y = 0 ?A. Y = E 2 X Y=e^{2 X} Y = E 2 X B. Y = 2 E X Y=2 E^x Y = 2 E X C. Y = Sin ⁡ ( 2 X Y=\sin (2 X Y = Sin ( 2 X ]D. Y = Cos ⁡ ( 2 X Y=\cos (2 X Y = Cos ( 2 X ]

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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. A differential equation is a mathematical equation that involves an unknown function and its derivatives. In this article, we will focus on solving a specific type of differential equation, namely the second-order linear homogeneous differential equation.

What is a Second-Order Linear Homogeneous Differential Equation?

A second-order linear homogeneous differential equation is a differential equation of the form:

ay′′+by′+cy=0ay^{\prime \prime} + by^{\prime} + cy = 0

where aa, bb, and cc are constants, and yy is the unknown function. In this equation, y′y^{\prime} represents the first derivative of yy, and y′′y^{\prime \prime} represents the second derivative of yy.

The Given Differential Equation

The given differential equation is:

1y′′−4y=01 y^{\prime \prime}-4 y=0

This is a second-order linear homogeneous differential equation with a=1a=1, b=0b=0, and c=−4c=-4.

Solving the Differential Equation

To solve this differential equation, we can use the characteristic equation method. The characteristic equation is obtained by substituting y=erxy=e^{rx} into the differential equation, where rr is a constant.

1r2−4=01 r^2-4=0

Solving for rr, we get:

r2=4r^2=4

r=±2r=\pm 2

The general solution of the differential equation is:

y=c1e2x+c2e−2xy=c_1 e^{2 x}+c_2 e^{-2 x}

where c1c_1 and c2c_2 are arbitrary constants.

Evaluating the Options

Now, let's evaluate the options given in the problem:

A. y=e2xy=e^{2 x}

This is a solution to the differential equation, as it satisfies the equation:

1(2)2−4(e2x)=01 (2)^2-4 (e^{2 x})=0

B. y=2exy=2 e^x

This is not a solution to the differential equation, as it does not satisfy the equation:

1(2)2−4(2ex)=01 (2)^2-4 (2 e^x)=0

C. y=sin⁡(2x)y=\sin (2 x)

This is not a solution to the differential equation, as it does not satisfy the equation:

1(−4sin⁡(2x))−4sin⁡(2x)=01 (-4 \sin (2 x)) - 4 \sin (2 x) = 0

D. y=cos⁡(2x)y=\cos (2 x)

This is not a solution to the differential equation, as it does not satisfy the equation:

1(−4cos⁡(2x))−4cos⁡(2x)=01 (-4 \cos (2 x)) - 4 \cos (2 x) = 0

Conclusion

In conclusion, the correct solution to the differential equation 1y′′−4y=01 y^{\prime \prime}-4 y=0 is:

y=c1e2x+c2e−2xy=c_1 e^{2 x}+c_2 e^{-2 x}

This solution satisfies the differential equation and is a valid solution.

Final Answer

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 our previous article, we discussed how to solve a specific type of differential equation, namely the second-order linear homogeneous differential equation. In this article, we will answer some frequently asked questions (FAQs) on differential equations.

Q: What is a differential equation?

A: A differential equation is a mathematical equation that involves an unknown function and its derivatives. It is a fundamental concept in mathematics and is used to model various phenomena in physics, engineering, and economics.

Q: What are the different types of differential equations?

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

  • First-order differential equations: These are differential equations that involve the first derivative of the unknown function.
  • Second-order differential equations: These are differential equations that involve the second derivative of the unknown function.
  • Linear differential equations: These are differential equations that can be written in the form ay′′+by′+cy=0ay^{\prime \prime} + by^{\prime} + cy = 0, where aa, bb, and cc are constants.
  • Nonlinear differential equations: These are differential equations that cannot be written in the form ay′′+by′+cy=0ay^{\prime \prime} + by^{\prime} + cy = 0, where aa, bb, and cc are constants.

Q: How do I solve a differential equation?

A: There are several methods to solve a differential equation, including:

  • Separation of variables: This method involves separating the variables in the differential equation and integrating both sides.
  • Integration factor: This method involves multiplying both sides of the differential equation by an integration factor to simplify the equation.
  • Characteristic equation: This method involves substituting y=erxy=e^{rx} into the differential equation and solving for rr.

Q: What is the characteristic equation?

A: The characteristic equation is a polynomial equation that is obtained by substituting y=erxy=e^{rx} into the differential equation. It is used to find the values of rr that satisfy the differential equation.

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 find the particular solutions that satisfy the differential equation. The general solution is a linear combination of the particular solutions.

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

A: A particular solution is a specific solution that satisfies the differential equation, while a general solution is a linear combination of all possible solutions that satisfy the differential equation.

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 several software packages available, such as MATLAB and Mathematica, that can be used to solve differential equations.

Q: What are some common applications of differential equations?

A: Differential equations have many applications in various fields, including:

  • Physics: Differential equations are used to model the motion of objects, the behavior of electrical circuits, and the behavior of mechanical systems.
  • Engineering: Differential equations are used to model the behavior of electrical circuits, mechanical systems, and thermal systems.
  • Economics: Differential equations are used to model the behavior of economic systems, including the behavior of interest rates and the behavior of stock prices.

Conclusion

In conclusion, differential equations are a fundamental concept in mathematics and have many applications in various fields. We hope that this article has provided you with a better understanding of differential equations and how to solve them.

Final Answer

There is no final answer to this article, as it is a collection of FAQs on differential equations. However, we hope that this article has provided you with a better understanding of differential equations and how to solve them.