Which Of The Following Is A Solution Of The Differential Equation 3 Y ′ ′ + 3 Y = 0 3 Y^{\prime \prime}+3 Y=0 3 Y ′′ + 3 Y = 0 ?A. E − 5 X + E X E^{-5 X}+e^x E − 5 X + E X B. C 1 Cos ⁡ X + C 2 Sin ⁡ X C_1 \cos X+C_2 \sin X C 1 ​ Cos X + C 2 ​ Sin X

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 homogeneous linear differential equation with constant coefficients.

What is a Homogeneous Linear Differential Equation with Constant Coefficients?

A homogeneous linear differential equation with constant coefficients is a differential equation of the form:

$$a_n y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_1 y^{\prime} + a_0 y = 0$$

where $a_n, a_{n-1}, \dots, a_1, a_0$ are constants, and $y$ is the unknown function. The order of the differential equation is $n$, which is the highest order of the derivative of $y$.

The Given Differential Equation

The given differential equation is:

$$3 y^{\prime \prime} + 3 y = 0$$

This is a second-order homogeneous linear differential equation with constant coefficients. The coefficient of the second derivative is $3$, and

Q&A: Solving the Differential Equation

Q: What is the general solution of the differential equation $3 y^{\prime \prime} + 3 y = 0$?

A: The general solution of the differential equation $3 y^{\prime \prime} + 3 y = 0$ is given by:

$$y = C_1 \cos x + C_2 \sin x$$

where $C_1$ and $C_2$ are arbitrary constants.

Q: How do we find the general solution of a homogeneous linear differential equation with constant coefficients?

A: To find the general solution of a homogeneous linear differential equation with constant coefficients, we need to find the roots of the characteristic equation. The characteristic equation is obtained by substituting $y = e^{rx}$ into the differential equation, where $r$ is a constant.

Q: What is the characteristic equation of the differential equation $3 y^{\prime \prime} + 3 y = 0$?

A: The characteristic equation of the differential equation $3 y^{\prime \prime} + 3 y = 0$ is given by:

$$3 r^2 + 3 = 0$$

Q: How do we solve the characteristic equation?

A: To solve the characteristic equation, we can use the quadratic formula:

$$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a = 3$, $b = 0$, and $c = 3$.

Q: What are the roots of the characteristic equation?

A: The roots of the characteristic equation are given by:

$$r = \pm i \sqrt{3}$$

Q: How do we use the roots of the characteristic equation to find the general solution?

A: The general solution of the differential equation is given by:

$$y = C_1 e^{r_1 x} + C_2 e^{r_2 x}$$

where $r_1$ and $r_2$ are the roots of the characteristic equation.

Q: What is the final answer to the problem?

A: The final answer to the problem is:

$$y = C_1 \cos x + C_2 \sin x$$

This is the general solution of the differential equation $3 y^{\prime \prime} + 3 y = 0$.

Conclusion

In this article, we have discussed how to solve a homogeneous linear differential equation with constant coefficients. We have used the characteristic equation to find the roots, and then used the roots to find the general solution. The final answer is $y = C_1 \cos x + C_2 \sin x$, which is the general solution of the differential equation $3 y^{\prime \prime} + 3 y = 0$.

Frequently Asked Questions

• Q: What is the order of the differential equation? A: The order of the differential equation is 2.
• Q: What is the coefficient of the second derivative? A: The coefficient of the second derivative is 3.
• Q: What is the general solution of the differential equation? A: The general solution of the differential equation is $y = C_1 \cos x + C_2 \sin x$.

References

• [1]: "Differential Equations" by Michael D. Greenberg
• [2]: "Introduction to Differential Equations" by James R. Brannan
• [3]: "Differential Equations and Dynamical Systems" by Lawrence Perko