Solving mx'' + cx' + kx = 0 Using Laplace

[morbidkokos]

Hello , can anyone help me solve this using laplace ?

mx'' + cx' + kx = 0 , x(0)=0 , x'(0)=0 , m > 0 , c > 0 , k > 0 and c^2-4km > 0


thanks in advance for your answer.(first time using this forum)

 

[Jasso]

Usually, we like to see what efforts have been put into a solution. It helps us know where you are stuck; it also makes sure we aren't doing all the work for you, which would make the learning process more difficult.

So please, show what you've done towards a solution so far.

 

[morbidkokos]

Hello again and thanks for your answer .
This is what I got so far :

mx''+cx'+kx=0 →
m[s^2 X(s) + sx(0) -x'(0)] + c[sX(s) - x(0)] + kX(s)=0 → $x(0)=0 , x'(0)=0$ →
X(s)(ms^2+cs+k)=0 →
since c^2-4km > 0 there are 2 roots a1 and a2.
Therefore X(s)(s-a1)(s-a2)=0

and that's where I'm stuck. I know that X(s)=0 is a solution but isn't there anything else?
(going to sleep now 3:10 am is too late for D.E.)

 

[HallsofIvy]

Because the coefficient of x'' is not zero, this problem has a unique solution.
It is obvious that x identically equal to 0 satisfies this equation so that is the solution.