Solution for a second-order differential equation

[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~

 

[BvU]

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

 

[chaksome]

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～

 

[BvU]

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

 

[HallsofIvy]

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.