For the harmonic oscillator system x\" + bx\' + kx =0 , find all values of b and

Question

For the harmonic oscillator system       x" + bx' + kx =0   , find all values of b and k for which this system has real,distinct eigenvalues. Find the general solution of this system in these cases. Find the solution of the system that satisfies the initialcondition (0,1). Describe the motion of the mass in this particular case. For the harmonic oscillator system       x" + bx' + kx =0   , find all values of b and k for which this system has real,distinct eigenvalues. Find the general solution of this system in these cases. Find the solution of the system that satisfies the initialcondition (0,1). Describe the motion of the mass in this particular case.

Explanation / Answer

For the given differential equation , the discriminant isgiven by, B2-4AC=b2-4(1)k=b2-4kis zero     --------> same eigen values andare real                                                i.e the possible values of (b,k) are (±2c,c) where c R                                           is postive --------> distinct and real                                              possible values of (b,k) are (b,c) where b (-,-2c)U(2c,) and c R                                           is negative ---------> distinct and imaginary                                               possible values of (b,k) are (b,c) where b (-2c, 2c) and c R Kindly , post the remaining bits of the question as separated problems in QA board END

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