A Typical Application In case the differential equation models a physical system, the nonhomogenous term F(x)frequently corresponds to some external influence on the system. Suppose that a mass m is attached both to a spring that exerts on it a force Fs and to a dashpot (shock absorber)that exerts a force FR on the mass.Assume that the restoring force Fs of the spring is proportional to the displacement x(positive to the right,negative to the left)of the mass from equilibrium, and that the dashpot force FR is proportional to the velocity v=dx/dt of the mass
A Typical Application In case the differential equation models a physical system, the nonhomogenous term F(x) frequently corresponds to some external influence on the system. Suppose that a mass m is attached both to a spring that exerts on it a force FS and to a dashpot (shock absorber) that exerts a force FR on the mass. Assume that the restoring force FS of the spring is proportional to the displacement x (positive to the right, negative to the left) of the mass from equilibrium, and that the dashpot force FR is proportional to the velocity v = dx/dt of the mass. Then we have FS = −kx and FR = −cv (k, c > 0)
A Typical Application In case the differential equation models a physical system, the nonhomogenous term F(x)frequently corresponds to some external influence on the system. Suppose that a mass m is attached both to a spring that exerts on it a force Fs and to a dashpot (shock absorber)that exerts a force FR on the mass.Assume that the restoring force Fs of the spring is proportional to the displacement x(positive to the right,negative to the left)of the mass from equilibrium, and that the dashpot force FR is proportional to the velocity v=dx/dt of the mass.Then we have Fs =-kx and FR=-cv (k,c>0)
A Typical Application In case the differential equation models a physical system, the nonhomogenous term F(x) frequently corresponds to some external influence on the system. Suppose that a mass m is attached both to a spring that exerts on it a force FS and to a dashpot (shock absorber) that exerts a force FR on the mass. Assume that the restoring force FS of the spring is proportional to the displacement x (positive to the right, negative to the left) of the mass from equilibrium, and that the dashpot force FR is proportional to the velocity v = dx/dt of the mass. Then we have FS = −kx and FR = −cv (k, c > 0)
A Typical Application Newton's law F=ma now gives mx"=Fs+FR; (2) that is d2x dx m +kx=0 (3) 4口y4元卡2月只0
A Typical Application Newton’s law F = ma now gives mx 00 = FS +FR; (2) that is m d 2 x dt2 +c dx dt +kx = 0. (3) Thus we have a differential equation satisfied by the position function x(t) of the mass m. This homogeneous second-order linear equation governs the free vibrations of the mass
A Typical Application Newton's law F=ma now gives mx"=Fs+FR; (2) that is d2x dx +kx=0 (3) Thus we have a differential equation satisfied by the position function x(t)of the mass m.This homogeneous second-order linear equation governs the free vibrations of the mass. 4口1日，4t元3000
A Typical Application Newton’s law F = ma now gives mx 00 = FS +FR; (2) that is m d 2 x dt2 +c dx dt +kx = 0. (3) Thus we have a differential equation satisfied by the position function x(t) of the mass m. This homogeneous second-order linear equation governs the free vibrations of the mass
A Typical Application If,in addition to Fs and FR,the mass m is acted on by an external force F(t)-which must then be added to the right- hand side in(2)-the resulting equation is 4日1①y至，无2000
A Typical Application If, in addition to FS and FR, the mass m is acted on by an external force F(t)− which must then be added to the righthand side in (2)-the resulting equation is m d 2 x dt 2 +c dx dt +kx = F(t). (4) This nonhomogeneous linear differential equation governs the forced vibrations of the mass under the influence of the external force F(t)