Second-order linear differential equations

Abstract

A second-order differential equation is an equation of the form

$${d^2y\over dt^2} = f\Bigg(t,y, {dy\over dt}\Bigg).$$

((1))

For example, the equation

$${d^2y\over dt^2}={\rm sin}t+3y+\Big({dy \over dt}\Big)^2$$

is a second-order differential equation. A function y = y(t) is a solution of (1) if y(t) satisfies the differential equation; that is

$${d^2y(t)\over dt^2} = f\Bigg(t,y(t), {dy(t)\over dt}\Bigg).$$

Thus, the function y(t) = cost is a solution of the second-order equation d ² y/dt ²= −y since d ²(cos t)/dt ²= −cost.