Systems of First-Order Differential Equations
- Consider the system initial value problem
(29a)
(29b)
I
Y1
1
Y1 - 2Y1Y2 + --
x + 2
Y~ Y1 + Y2 + y~ - tan x
Y1 (0) = 1, Y2(0) = 2.
a. Is the system (29a) linear or nonlinear?
331
b. Applying the appropriate theorem from this chapter, what can be
said about the interval on which a unique solution to this problem exists?
- Consider the system initial value problem
(30a)
I
Y1
I
Y2
(30b) Y1(0)=0 Y2(0)=1.
a. Is the system (30a) linear or nonlinear?
b. Applying the appropriate theorem from this chapter, what can be
said about the interval on which a unique solution to this problem exists?
c. Show that {y 1 (x) = x, Y2(x) =ex} is the solution to the IVP (30).
On what interval is this the solution to the initial value problem?
- Consider the system initial value problem
(31a)
(31b)
I
Y1 _Jj_}:_ + _Jj2._
2-y2 x+3
-----Y2 Y1
2 + Y1 x - 4
Y1(0) = 1, Y2(0) = 1.
a. Is system (31a) linear or nonlinear?
b. What can be said about the interval on which a unique solution to
this problem exists?
c. Analyze this initial value problem and complete the following state-
ment. The interval of existence and uniqueness will terminate at the
point x = a if any of the following occurs as x approaches a, x __, __ ,
x __, _ , Y1(x) __, _ , Y1(x) __, _, Y2(x) __, _, Y2(x) __, _.
