Linear Differential Equations 1. Linear Differential Equations A linear differential equation is of the following form: . a n x ( n ) + a n −1 x ( n −1) + · · · + a 1 x + a 0 x = q ( t ). (1) The ak ’s are the coefﬁcients. They may depend upon t (but not on x ). If an is not zero then the differential equation is said to be of order n. If this models a physical system then the left hand side represents the system and the right hand side represents the input signal. The coefﬁcients represent parameters of the system. For example, the mass, damping and .. . spring constants m, b and k in mx + bx + kx are the parameters of the sys tem. In general, they may depend on time, e.g. maybe the force is actually a rocket, and the fuel burns so m decreases. Or maybe the spring gets softer as it ages. Maybe the honey in the dashpot gets thicker with time. We will generally assume the coefﬁcients are constant. In which case equation (1) is said to be a constant coefﬁcient linear equation. It is, in fact, a good approximation of the non-constant coefﬁcient equation as long as the coefﬁcients vary on a time-scale that is much greater than the timescale of the dynamical variable x. 2. Second Order Homogeneous Constant Coefﬁcient Linear Equations .. . We will study the spring system mx + bx + kx = Fext starting with the case Fext = 0. .. . mx + bx + kx = 0. (2) To ensure that (2) is of second order (and a realistic physical system) we always assume m > 0, but we will allow the case b = 0 and occasionally k = 0. With no external force the system evolves on its own. Think of a door that can swing back and forth or a ball on the end of a rubber band. As we did in ﬁrst order equations we will call (2) a homogeneous linear differential equation. 3. The Undamped Case The special case b = 0 is called undamped. This is called the simple .. harmonic oscillator. It’s ODE is mx + kx = 0 or .. x+ k x = 0. m If we let ω = √ k/m our equation becomes .. x + ω 2 x = 0. We have seen before (and you can easily check) that x1 (t) = cos(ωt) and x2 (t) = sin(ωt) are solutions to this equation. Since the input is 0 and the equation is linear, we can use superposition of solutions to get the general solution x (t) = a cos(ωt) + b sin(ωt) = A cos(ωt − φ) (3) This is another fundamental fact you should memorize! (The second equality comes from the sinusoidal identity, which gives a = A cos φ and b = A sin φ.) . We know (3) gives every solution because x (0) = a and x (0) = ωb, so you can solve (uniquely) for a and b to give any desired intial condition.