Second-order homogeneous linear differential equation

In summary, the second-order homogeneous linear differential equation with a coefficient of $K = 4$ has a repeated root of $r = -2$, resulting in a general solution of $y = C_1e^{-2t} + C_2 te^{-2t}$. The form of the general solution can be proven using the reduction of order method.
  • #1
shamieh
539
0
Consider the second-order homogeneous linear differential equation $y'' + 4y' + Ky = 0$

Find the general solution if $K = 4$

So here is what I have:

$r^2 + 4r + 4 = 0 $
=$(r + 2)(r+2)$
$r=-2$ ?

But I thought that you can't do this because you won't be learning anything new if you have two of the same solutions. I'm not sure what to do
 
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  • #2
If you have a repeated root $r$, then the general solution is:

\(\displaystyle y(x)=c_1e^{rx}+c_2xe^{rx}\)
 
  • #3
so $y = C_1e^{-2t} + C_2 te^{-2t}$
 
  • #4
Yes...as an exercise you may wish to use the reduction of order method to prove the form of the general solution is as I gave above in the case of a repeated root for the characteristic equation. :D
 

FAQ: Second-order homogeneous linear differential equation

What is a second-order homogeneous linear differential equation?

A second-order homogeneous linear differential equation is a mathematical equation that describes a relationship between a function and its derivatives, where the highest derivative present is of second order (meaning it is raised to the power of 2), and there are no non-homogeneous terms (meaning the function and its derivatives only appear on one side of the equation). The general form of a second-order homogeneous linear differential equation is:
y'' + p(x)y' + q(x)y = 0

What are the key characteristics of a second-order homogeneous linear differential equation?

A second-order homogeneous linear differential equation has three key characteristics: it is a linear equation, meaning the function and its derivatives appear only in a linear form; it is homogeneous, meaning there are no non-homogeneous terms present; and it is of second order, meaning the highest derivative present is of second order. These characteristics are important in understanding how to solve these types of equations.

What is the general solution to a second-order homogeneous linear differential equation?

The general solution to a second-order homogeneous linear differential equation is a function that satisfies the equation for all possible values of the independent variable. It is typically written in the form of y(x) = c1y1(x) + c2y2(x), where c1 and c2 are arbitrary constants, and y1(x) and y2(x) are two linearly independent solutions to the differential equation. This general solution can be derived using techniques such as the method of undetermined coefficients or the method of variation of parameters.

How do you solve a second-order homogeneous linear differential equation with constant coefficients?

To solve a second-order homogeneous linear differential equation with constant coefficients, you can use the characteristic equation method. This involves finding the roots of the characteristic equation, which is obtained by replacing the coefficients of the differential equation with constants. Depending on the nature of the roots (real, complex, repeated), you can then determine the form of the general solution and solve for the arbitrary constants using initial conditions.

How is a second-order homogeneous linear differential equation used in science?

Second-order homogeneous linear differential equations are used in various fields of science, such as physics, engineering, and economics, to model natural phenomena and make predictions. For example, the motion of a simple pendulum can be described by a second-order homogeneous linear differential equation, and the solution to this equation can be used to calculate the period and amplitude of the pendulum's oscillations. In engineering, these equations are used to model systems such as electrical circuits and mechanical systems, allowing engineers to design and optimize their performance.

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