How Is Displacement Solved in This Differential Equation?

[LeDragonian]

It has been a while since I was involved with my differential equations. I am a mech student. I was trying out a sample problem from the dynamics book and came upon this equation.

k = some constant of proportionality for a spring pushing back a spring mounted slider.

(1/k) arcsin(ks/v_0) = t --> becomes s = (v_0/k) sin(kt)

which is an expression for displacement. I want to know how it is solved for displacement s.

Thank you.

 

[Quarlep]

If you have an equation like ##arcsint=a## you can write it ##sina=t## here
(1/k) arcsin(ks/v₀)=t
arcsin(ks/v₀)=tk
then here apply what I did then we get
sin(tk)=ks/v₀
then simply s is ;
sin(tk)v₀/k=s

 

[LeDragonian]

If you have an equation like ##arcsint=a## you can write it ##sina=t## here
(1/k) arcsin(ks/v₀)=t
arcsin(ks/v₀)=tk
then here apply what I did then we get
sin(tk)=ks/v₀
then simply s is ;
sin(tk)v₀/k=s

Thank you. It reminded me of when I solve for theta in some other equations and it rightly should make sense.