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 d2x dx +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). 4口0y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 dt2 +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)
Homogeneous Second-Order Equations For general second-order linear equation A(x)y"+B(x)y+C(x)y=F(x) A,B,C,F are continuous on interval I.Assume that A(x)0 at each point of / y”+p(xy+q(xy=f(x) (5) The associated homogenous equation is thus y”+p(x)y+q(xy=0 (6) 4口y4元卡B月只0
Homogeneous Second-Order Equations For general second-order linear equation A(x)y 00 +B(x)y 0 +C(x)y = F(x) A,B,C,F are continuous on interval I. Assume that A(x) 6= 0 at each point of I. y 00 +p(x)y 0 +q(x)y = f(x) (5) The associated homogenous equation is thus y 00 +p(x)y 0 +q(x)y = 0 (6)
y"+p(x)y'+q(x)y =0 (6) Theorem(Superposition for Homogeneous Equations) Let y1,y2 be two solutions,c1,c2 are two constants,then y=ciy1+c2y2 is also a solution. 4日10y至，无2000
y 00 +p(x)y 0 +q(x)y = 0 (6) Theorem (Superposition for Homogeneous Equations) Let y1, y2 be two solutions, c1, c2 are two constants, then y = c1y1 +c2y2 is also a solution
Example We see that y1(x)=cosx and y2(x)=sinx are two solutions of y"+y=0 (7) From the Theorem,any y(x)=cI cosx+c2sinx (8) is solution to(7).Conversely,any solution of(7)can be written as(8). 4口14①y至元2000
Example We see that y1(x) = cos x and y2(x) = sinx are two solutions of y 00 +y = 0 (7) From the Theorem, any y(x) = c1 cos x+c2 sinx (8) is solution to (7). Conversely, any solution of (7) can be written as (8)
Existence and Uniqueness for Linear Equations Theorem Suppose that p,q,f are continuous on the open interval I con- taining the point a.Then, y”+p(xy+q(xy=fx) (9) has a unique solution on the entire interval I that satisfies the initial conditions y(a)=bo;y(a)=b1. 4口卡0y至，无，2000
Existence and Uniqueness for Linear Equations Theorem Suppose that p,q,f are continuous on the open interval I containing the point a. Then, y 00 +p(x)y 0 +q(x)y = f(x) (9) has a unique solution on the entire interval I that satisfies the initial conditions y(a) = b0, y 0 (a) = b1