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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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 $$\lambda^2\pm\lambda+(\pm 1+k)=0$$ giving $$\lambda_{1,2}=\frac{\mp 1\pm\sqrt{1-4(\pm 1+k))}}{2}$$ and depending on the value of $k$ you will get different sets of solutions. |
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