x=A cos nt+B sin nt;(d^(2)x)/(dt^(2))=?

If x=a cos nt-b sin nt and (d^(2)x)/(dt^(2))=lambda x then find the value of lambda.

If x=A cos 4t+B sin4t, then (d^(2)x)/(dt^(2)) is equal to

If x=A cos 4t+B sin4t, then (d^(2)x)/(dt^(2)) is equal to

If x=a cos nt-b sin nt, then (d^(2)x)/(dt^(2)) is n^(2)x(b)-n^(2)x(c)-nx(d)nx

If x=a cos nt-b sin nt, then (d^(2)x)/(dt^(2)) is n^(2)x( b) -n^(2)x( c) nx( d )nx

If x=a cos nt-b sin nt , then (d^2x)/(dt^2)=

If x= A cos 4t + B sin 4t , then (d^(2)x)/(dt^(2))=

If x=a (cos t + t sin t) and y = a (sin t - t cos t ) , find (d^2x)/(dt^2),(d^2y)/(dt^2) and (d^2y)/(dx^2) . Also mention the domain of validity.

Show that , the solution x = e^(-kt) (a cos nt + b sin nt ), for all a and b , always satisfies the differenital equation (d^(2)x)/(dt^(2)) + 2k (dx)/(dt) + (k^(2) + n^(2)) x = 0