This will be the same differential equation as with external force application, but displacement corresponds there to initial conditions for one or several variables. Another case, when displacement is limited by analytical or logical conditions, this is a case of non-linear equation.

t is time X is displacement, X0(t) is the imposed displacement, F0(t) imposed force

M(d2(X-X0)/dt2) +C (d(X-X0)/dt)+K(X-X0)=F0(t).

or F0(t)+KX0(t)+M(d2(X0)/dt2)(t)+C(d(X0)/dt)(t) is the force imposed

It is a very interested question ,see

http://www.academia.edu/33655682/Theory_of_Vibration_with_Applications_4th_Edition_-_By_William_T._Thomson_

Thank you Juma, but where in the book?

Thank you @Pierre I got it