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k
1
(spring)
k
2
(spring)
d
2
(damp)
d
1
(damp)
Problem 6.20. Write down the following higher order equations in the form of a system of first
order equations. If an equation is linear, write it in the matrix form.
i. y
000
+ xy
00
+ 2y = 0
ii. y
000
+ sin(y) = y
0
¡1
Problem 6.21. Assume that F (y) is a smooth vector field for y 2 R
n
. Prove that it is impossible
that two trajectories of the system y
0
= F (y) intersect each other.
Problem 6.22. We observed that vector fields are tangent to the associated traj ectori es. From the
physics point of view, the vector fi eld of a first-order system is the velocity fi eld of particle moving in
the phase plane. This property can be used to construct the trajectories numerically as well. Assume
a particle is located at (x
0
; y
0
) in the phase plane at time t = 0 following the equation
d
dt
x
y
= F (x; y):
The velocity vector at (x
0
; y
0
) is v~ = F (x
0
; y
0
). Therefore, on e can use the linear approximation for-
mula at t = h 1 and write
x(h)
y(h)
≈
x
0
y
0
+ hF (x
0
; y
0
):
Accordingly, we obtain the following recursive formula
x
n+1
y
n+1
≈
x
n
y
n
+ hF (x
n
; y
n
);
where x
n
:= x(nh), y
n
:= y(nh) for small step size h.
a) Follow the above algorithm and draw the solution x(t); y(t) o f the system (6.1) for t 2 [0; 2]
using the step size h = 0.1, and the initial condition x
0
= 8; y
0
= 5. Take parameters as r
1
= 4;
k
1
= 0.4; r
2
= 2; k
2
= 0.2.
b) Draw the obtained solution in the phase plane (x; y).
6.3 Linear homogeneous systems
We present a method to solve 2D systems wi th constant coefficient matrices, y
0
= A y,
Y 2R
2
. There is no general method to solve l inear s ystems with variable coefficients.
6.3.1 Outline of the m e t hod
Assume that a particle is moving in the phase plane (y
1
; y
2
) subj ect to the law y
0
= Ay,
and assume that v~ is an eigenvector o f A with eigenvalue λ. Consider the following initial
value problem
(
d
dt
y = Ay
y(0) = v~
:
Note that
dy
dt
(0) = Ay(0) = A v~ = λ v~ ;
16 Systems of Differential Equations