Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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
add comment

1 Answer

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

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

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.