Need solution step by step Thanks 359 Differential equation

Need solution step by step

Thanks

3.5.9 Differential equation of motion of a particle is 3dkldt\'+12dxdttex 0. Determine the maximal value of stiffness c when motion will be aperiodic,

Solution

3d^2x/dt^2 + 12 dx/dt + cx=0

which can be written as

3m^2+12m+c=0

if we have to get two roots

then 12^2-4x3xc >0

so, 12(12-c)>0

so, 12-c >0

or c<12