# Solution for a second-order differential equation

• chaksome
In summary, the conversation discusses the method of using Laplace transforms to work out x(t), regardless of the form of f(t). The method involves using the associated homogeneous equation and the characteristic equation to find the general solution, and then using variation of parameters to find a solution to the entire equation. This method is explained in more detail in the provided link, and the calculations can be tedious but manageable.
chaksome
Homework Statement
Second order ODE
Relevant Equations
x‘’(t)+4x’(t)+8x(t) = f(t)
I wish to know if there is a method to work out x(t).
[No matter which form f(t) is]
Thank you~

You know PF: what's your attempt ? Know about Laplace transforms ?

BvU said:
You know PF: what's your attempt ? Know about Laplace transforms ?
Oh， I haven‘t learned that^v^
Do you mean this chapter will help me understand this problem？
I can try to learn it～

It gives you a formal, general method to find a solution, yes.

First, the "associated homogenous equation" is x''+ 8x'+ 4x= 0 which has "characteristic equation" r^2+ 8r+ 4= 0. Solve that by "completing the square": r^2+ 8r+ 16- 16+ 4= (r+ 4)^2- 12= 0 so (r+ 4)^2= 12, $r= -4\pm2\sqrt{3}$. The general solution to the "associate homogeneous equation" is $e^{-4x}\left(Ae^{2x\sqrt{3}}+ Be^{-2x\sqrt{3}}\right)$.

Now use "variation of parameters". We look for a solution to the entire equation of the form $y(x)= e^{-4x}\left(u(x)e^{2x\sqrt{3}}+ v(x)e^{-2x\sqrt{3}}\right)$. That is, we let the "parameters", A and B, vary. Surely That method is given in your textbook. You can also read about it at http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx.

The calculations are tedious but doable! The result will be an integral in f(x) that you add to the general solution to the homogeneous equation.

chaksome

## 1. What is a second-order differential equation?

A second-order differential equation is a mathematical equation that relates the second derivative of a function to the function itself. It is commonly used to model physical systems in fields such as physics, engineering, and economics.

## 2. How do you solve a second-order differential equation?

The general method for solving a second-order differential equation involves finding a particular solution that satisfies the equation, and then combining it with the complementary solution, which is the solution to the associated homogeneous equation. This can be done using various techniques such as separation of variables, substitution, or using a series solution.

## 3. What is the difference between an initial value problem and a boundary value problem for a second-order differential equation?

An initial value problem for a second-order differential equation involves finding a solution that satisfies the equation and a set of initial conditions, usually in the form of initial values for the function and its first derivative. A boundary value problem, on the other hand, involves finding a solution that satisfies the equation and a set of boundary conditions, usually at two or more distinct points.

## 4. Can a second-order differential equation have multiple solutions?

Yes, a second-order differential equation can have multiple solutions. This depends on the specific equation and the initial or boundary conditions given. In some cases, there may be an infinite number of solutions.

## 5. What are some real-world applications of second-order differential equations?

Second-order differential equations are commonly used in physics to model the motion of objects under the influence of forces, such as in the case of a pendulum or a mass-spring system. They are also used in engineering to analyze the behavior of mechanical systems, and in economics to model changes in economic variables over time.

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