Question

how the plot of solution of d²x/dt² = kx and d²x/dt² = - kx equations will differ ?

Answer 1

1)

This is a second order diffrential equation of the form

where   

The solution to this second order differential eqaution is given as,

---------------------------------------------- (1)

This solution means that the function is decaying that means

i) for   ,    wiill be infinity which is not possible therefore will be the solution.

ii)Similarly when , will be infinity therefore   will be the solution.

Hence for any case, the solution will always be exponentially decaying. Example of the plot is shown below:

2)   

,   ,

We know ,

The solution of such differential equation is given by,

-------------------------------------------------- (2)

We know,

Therefore for any value of t, the solution will always be either cos or sine function which is oscillatory. Example of the plot as a sine function solution is shown below.