同济大学：《常微分方程》课程教学资源（讲义）线性微分方程组 Linear Systems of Differential Equations（负责人：尚培培）

Linear of Differential Equations 4口14①yt至2000 Linear of Differential Equations

Linear Systems of Differential Equations Linear Systems of Differential Equations

Systems of Differential Equations So far,we have studied the differential equation that in- volves an unknown function and derivatives of this function. 4日0,1无2000 Linear of Differential Equations

Systems of Differential Equations So far, we have studied the differential equation that in￾volves an unknown function and derivatives of this function. In this chapter we shall deal with sets of simultaneous equations that involve several unknown functions and their derivatives. Example    d 2 x1 dt 2 −2 dx1 dt − dx2 dt = −3e 3t , dx2 dt −6x1 = 0. Linear Systems of Differential Equations

Systems of Differential Equations So far,we have studied the differential equation that in- volves an unknown function and derivatives of this function.In this chapter we shall deal with sets of simultaneous equations that involve several unknown functions and their derivatives. 4日10,1无2000 Linear of Differential Equations

Systems of Differential Equations So far, we have studied the differential equation that in￾volves an unknown function and derivatives of this function. In this chapter we shall deal with sets of simultaneous equations that involve several unknown functions and their derivatives. Example    d 2 x1 dt 2 −2 dx1 dt − dx2 dt = −3e 3t , dx2 dt −6x1 = 0. Linear Systems of Differential Equations

Systems of Differential Equations So far,we have studied the differential equation that in- volves an unknown function and derivatives of this function.In this chapter we shall deal with sets of simultaneous equations that involve several unknown functions and their derivatives. Example x1_2d_de=-3e, 2 dt dt -6x1=0. 4口14①y至元2000 Linear of Differential Equations

Systems of Differential Equations So far, we have studied the differential equation that in￾volves an unknown function and derivatives of this function. In this chapter we shall deal with sets of simultaneous equations that involve several unknown functions and their derivatives. Example    d 2 x1 dt2 −2 dx1 dt − dx2 dt = −3e 3t , dx2 dt −6x1 = 0. Linear Systems of Differential Equations

Example Consider the system of two masses and two springs shown in Fig5.1.1,with a given external force f(r)acting on the right- hand mass m2.We denote by x(t)the displacement (to the right)of the mass m from its static equilibrium position(when the system is motionless and in equilibrium and f()=0),and by y(r)the displacement of the mass m2 from its static position. Thus the two springs are neither stretched nor compressed when x and y are zero. 4口14①yt至2000 Linear of Differential Equations

Example Consider the system of two masses and two springs shown in Fig5.1.1, with a given external force f(t) acting on the right￾hand mass m2. We denote by x(t) the displacement (to the right) of the mass m1 from its static equilibrium position (when the system is motionless and in equilibrium and f(t) = 0), and by y(t) the displacement of the mass m2 from its static position. Thus the two springs are neither stretched nor compressed when x and y are zero. Linear Systems of Differential Equations

Example In the configuration in Fig.5.1.1,the first spring is stretched x units and the second by y-x units. 4口1y1至，12000 Linear of Differential Equations

Example In the configuration in Fig.5.1.1, the first spring is stretched x units and the second by y−x units. We apply Newton’s law of motion to the two “free body diagrams” shown in Fig.5.1.2; we thereby obtain the system m1x 00 = −k1x+k2(y−x) (1) m2y 00 = −k2(y−x) +f(t) of differential equations that the position function x(t) and y(t) must satisfy. For instance, if m1 = 2,m2 = 1, k1 = 4, k2 = 2, and f(t) = 40 sin 3t in appropriate physical units, then the system in (1) reduces to 2x 00 = −6x+2y (2) y 00 = 2x−2y+40 sin 3t. Linear Systems of Differential Equations

Example In the configuration in Fig.5.1.1,the first spring is stretched x units and the second by y-x units.We apply Newton's law of motion to the two "free body diagrams"shown in Fig.5.1.2;we thereby obtain the system m1x”=-k1x+k20y-) (1) m2y”=-k20y-x)+ft) of differential equations that the position function x(t)and y(t) must satisfy. 4口10,1至，1无13000 Linear of Differential Equations

Example In the configuration in Fig.5.1.1, the first spring is stretched x units and the second by y−x units. We apply Newton’s law of motion to the two “free body diagrams” shown in Fig.5.1.2; we thereby obtain the system m1x 00 = −k1x+k2(y−x) (1) m2y 00 = −k2(y−x) +f(t) of differential equations that the position function x(t) and y(t) must satisfy. For instance, if m1 = 2,m2 = 1, k1 = 4, k2 = 2, and f(t) = 40 sin 3t in appropriate physical units, then the system in (1) reduces to 2x 00 = −6x+2y (2) y 00 = 2x−2y+40 sin 3t. Linear Systems of Differential Equations

Example In the configuration in Fig.5.1.1,the first spring is stretched x units and the second by y-x units.We apply Newton's law of motion to the two "free body diagrams"shown in Fig.5.1.2;we thereby obtain the system m1x”=-k1x+k2y-x) (1) m2y”=-k20y-x)+f(t) of differential equations that the position function x()and y(t) must satisfy.For instance,if m=2,m2=1,k1=4,k2 =2,and f(t)=40sin3t in appropriate physical units,then the system in (1)reduces to 2”=-6x+2y (2) y”=2x-2y+40sin3t. Linear of Differential Equations

Example In the configuration in Fig.5.1.1, the first spring is stretched x units and the second by y−x units. We apply Newton’s law of motion to the two “free body diagrams” shown in Fig.5.1.2; we thereby obtain the system m1x 00 = −k1x+k2(y−x) (1) m2y 00 = −k2(y−x) +f(t) of differential equations that the position function x(t) and y(t) must satisfy. For instance, if m1 = 2,m2 = 1, k1 = 4, k2 = 2, and f(t) = 40 sin 3t in appropriate physical units, then the system in (1) reduces to 2x 00 = −6x+2y (2) y 00 = 2x−2y+40 sin 3t. Linear Systems of Differential Equations

The Method of Elimination Example Find the particular solution of the system x=4x-3y,y=6x-7y (3) that satisfies the initial conditions x(0)=2,y(0)=-1. 4口14①yt至2000 Linear of Differential Equations

The Method of Elimination Example Find the particular solution of the system x 0 = 4x−3y, y 0 = 6x−7y (3) that satisfies the initial conditions x(0) = 2, y(0) = −1. Solution: If we solve the second equation in (3) for x, we get x = 1 6 y 0 + 7 6 y, (4) so that x 0 = 1 6 y 00 + 7 6 y 0 . (5) Linear Systems of Differential Equations

The Method of Elimination Example Find the particular solution of the system X=4x-3y,y=6x-7y (3) that satisfies the initial conditions x(0)=2,y(0)=-1 Solution:If we solve the second equation in(3)for x,we get (4) 4口10y至，元2000 Linear of Differential Equations

The Method of Elimination Example Find the particular solution of the system x 0 = 4x−3y, y 0 = 6x−7y (3) that satisfies the initial conditions x(0) = 2, y(0) = −1. Solution: If we solve the second equation in (3) for x, we get x = 1 6 y 0 + 7 6 y, (4) so that x 0 = 1 6 y 00 + 7 6 y 0 . (5) Linear Systems of Differential Equations