I'm trying to solve this differential equation but having some trouble as there is a constant in there which would change the solution depending on its value:

$$\ddot{x}+2k\dot{x}+10k^2x=0~~~~~~~(k>0),~(x(0)=0),~ (\dot{x}(0)=u).$$

I need to solve for $x$ in terms of $t$

I got down to trying to solve my characteristic equation which has roots:

$$\lambda_{1,2}=-k\pm\sqrt{k^2-10k}$$ but unsure where to go from here. Surely depending on $k$ this will have either real or complex roots in which case the solution will have a different form so I'm unsure how to proceed.

Any help?

  • 2
    $\begingroup$ I think you should have $\lambda_{1,2}=(-1\pm3i)k$. $\endgroup$ – mickep May 15 '15 at 8:44
  • $\begingroup$ You forgot that it is $4ac$ so that's $4\cdot 1\cdot 10k^2$ $\endgroup$ – Chinny84 May 15 '15 at 8:48
  • $\begingroup$ Thank you so much guys I was wondering what was going on turns out I multiplied by $c=10k$ not $10k^2$ silly mistake. $\endgroup$ – Carl May 15 '15 at 8:49

Hint: The caracteristc equation is: $$ \lambda^2+2k\lambda +10 k^2=0 $$ That has solutions: $$ \lambda_1=(-1-3i)k \qquad \lambda_1=(-1+3i)k $$

  • $\begingroup$ No square roots on the $3$. $\endgroup$ – mickep May 15 '15 at 8:48
  • 1
    $\begingroup$ Yes! stupid typo!!! thanks $\endgroup$ – Emilio Novati May 15 '15 at 8:49

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