The position y(t) of a damped spring satisfies the differential equation y''+2y'+5y=0. Determine y(t) when the initial position is y(0) and the initial velocity is y'(0)=1?
This type of differential equation is solved by substituting to get the characteristic equation . This equation has two roots that are complex conjugates, and , so the general solution is . You can use the initial conditions to find and .