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Linear Differential Equations: Solving (d^2+3d)y=0
Introduction
In this article, we will explore one of the most fundamental types of differential equations, known as linear differential equations. Specifically, we will focus on solving the differential equation (d^2+3d)y=0. This equation is a simple yet important example of a linear differential equation, and understanding its solution will provide a solid foundation for tackling more complex differential equations.
What is a Linear Differential Equation?
A linear differential equation is a differential equation in which the derivative of the unknown function is proportional to the function itself. In other words, a linear differential equation takes the form:
a Dy + by = f(x)
where a and b are constants, D is the derivative operator, and f(x) is a function of x.
The Equation (d^2+3d)y=0
The equation (d^2+3d)y=0 is a second-order linear homogeneous differential equation. The term "homogeneous" means that the equation does not have a constant term, and the term "second-order" means that the highest derivative of y is the second derivative (d^2y).
Solution to the Equation
To solve the equation (d^2+3d)y=0, we can use the method of undetermined coefficients. This method involves guessing a particular solution to the equation, and then using the equation to find the general solution.
Let's start by guessing a particular solution of the form:
y = Ae^(-3x)
where A is a constant.
Substituting this guess into the equation, we get:
(d^2+3d)(Ae^(-3x)) = 0
Expanding the derivative, we get:
(-3Ae^(-3x) + 9Ae^(-3x)) + 3(-3Ae^(-3x)) = 0
Simplifying, we get:
0 = 0
This shows that our guess is indeed a solution to the equation.
General Solution
Now that we have found a particular solution, we can use it to find the general solution. The general solution to a linear homogeneous differential equation is the sum of all possible particular solutions.
In this case, the general solution is:
y = Ae^(-3x) + Be^(-3x)
where A and B are arbitrary constants.
Interpretation of the Solution
The solution to the equation (d^2+3d)y=0 represents a family of curves that decay exponentially as x increases. The constants A and B determine the amplitude and phase shift of the curves, respectively.
For example, if A = 1 and B = 0, the solution is y = e^(-3x), which represents a curve that decays rapidly as x increases.
Conclusion
In this article, we have solved the linear differential equation (d^2+3d)y=0 using the method of undetermined coefficients. We have found the general solution to be y = Ae^(-3x) + Be^(-3x), where A and B are arbitrary constants. This solution represents a family of curves that decay exponentially as x increases.
Understanding the solution to this differential equation is important for modeling a wide range of phenomena in physics, engineering, and economics.
