Examples of Second Order ODE's
Mechanical Systems/Vibrations
Newton's second law asserts that the sum of the external forces acting on an object at any instant in time is equal to the product of the object's mass and acceleration. This has far reaching implications for ODE's.
Consider a mass attached to a spring that is allowed to move in a straight line in the horizontal direction. Let x(t), v(t), and a(t) denote the position, velocity, and acceleration, respectively, of the particle. We assume that x=0 is the equilibrium position of the spring. Hooke's law asserts that the force exerted on the mass by the spring is F(t)=-kx(t), where k is the spring constant. Newton's second law states: Mass times Acceleration = Sum of External Forces. Using the fact that a(t)=x''(t) we obtain the equation
This is a linear second-order ode. It is accompanied by the initial position x(0) and the initial velocity x'(0).
Notice that according to Hooke's law the force exerted by the spring depends linearly on the position x(t). In many applications the force depends nonlinearly on x(t). One example is Duffing's model. The differential equation is
Here e is a positive constant. This is an example of a nonlinear second-order ode.
Finally, suppose that there is damping in the spring-mass system. Damping might be provided by a dashpot that exerts a continuous force that is proportional to the velocity (F(t)=-cv(t), where c is a constant). The total force is a sum of force due to the spring and the damping. The differential equation is
Electrical Circuits
An RLC circuit consists of a resistor, an inductor, and capacitor in series with a voltage source. Let us assume that the resistance is R, the inductance is L, the capacitance is C, and the electromotive force is E(t). Let I(t) denote the current in the circuit. The current satisfies the differential equation:
This is an example of a linear second-order ode. To completely solve this problem the initial current I(0) and its derivative I'(0) must be specified.
One-Dimensional Free Fall Motion
Suppose that a particle initially at height 
is thrown
straight upward with velocity 
. Let s(t) denote the
particle's height, v(t) the particle's velocity, and a(t) the
objects acceleration at time t. If air resistance is neglected,
then by Newton's second law we have ma(t)=-mg. Using the
fact that a(t)=s''(t) and eliminating the mass, we obtain the
equation
The initial position s(0) and initial velocity s'(0)=v(0) must be specified. This is linear second-order ode.
Now suppose that air resistance is proportional to velocity.
This is another example of a linear second-order ode.