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After some algebraic simplification, I got the ODE: $$\ddot x(t)+\sqrt {(\dot x(t)+x(t))^2}+k x(t)=0$$ I interpreted this equation as: $$\ddot x(t)+|{(\dot x(t)+x(t))}|+k x(t)=0$$ I have some problem to solve it. Could you give me some hint please? Thanks.

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Normally when I have absolute values I split the problem into two cases. – Godisemo Mar 16 '12 at 8:40
You may use Godisemo's hint and then it becomes a 2nd ordered linear ODE with constant coefficients. – Tapu Mar 16 '12 at 9:01
up vote 1 down vote accepted

Your equation is equivalent to

$$\ddot x(t)\pm\dot x(t)\pm x(t)+kx(t)=0$$

so you have to solve the equations

$$\ddot x(t)+\dot x(t)+(1+k)x(t)=0 \qquad \ddot x(t)-\dot x(t)-(1-k)x(t)=0.$$

The solutions of these equations can be obtained by solving the characteristics equations

$$\lambda^2\pm\lambda+(\pm 1+k)=0$$


$$\lambda_{1,2}=\frac{\mp 1\pm\sqrt{1-4(\pm 1+k))}}{2}$$

and depending on the value of $k$ you will get different sets of solutions.

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