solving ordinary differential equations fig the effect of vaccination campaign oscillating one-dimensional systems numerous engineering constructions and devices contain materials that act like springs such springs give rise to oscillationsand controlling oscillations is key engineering task we shall now learn to simulate oscillating systems as alwayswe start with the simplest meaningful mathematical modelwhich for oscillations is second-order differential equationu tc td ( where is given physical parameter equation ( models one-dimensional system oscillating without damping ( with negligible dampingone-dimensional here means that some motion takes place along one dimension only in some coordinate system along with ( we need the two initial conditions and derivation of simple model many engineering systems undergo oscillationsand differential equations constitute the key tool to understandpredictand control the oscillations we start with the simplest possible model that captures the essential dynamics of an oscillating system some body with mass is attached to spring and moves along line without frictionsee fig for sketch (rolling wheels indicate "no friction"when the spring is stretched (or compressed)the spring force pulls (or pushesthe body back and work "againstthe motion more preciselylet tbe the position