1550078481-Ordinary_Differential_Equations__Roberts_

332 Ordinary Differential Equations

10. Consider the system initial value problem

(32a)

(32b)

2./x + 4Y1Y2 - (sinx)y2 + 3ex

-yr+ 4v5-xy1 -5ln7x

Y1 (2) = 3, Y2 (2) = -6.

a. Is the system (32a) 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) __. _.

11. Rewrite each of the following initial value problems as an equivalent
    first-order system initial value problem.
    a. y(4) = 3 xy2 + (yC1))3 exy(2)y(3) + x2 _ 1
    y(l) = -1, yCll(l) = 2, yC^2 l(1) = -3, yC^3 l(l) = 0

b. my" + cy' + k sin y = 0, where m, c, and k are positive constants

y(O) = 1, y' (0) = - 2

c. xy" - 3x^3 y' + (ln x )y = sin x

y(l) = -1, y'(l) = 0

d. (cos (x - y))yC^2 l - exy(^3 ) + xy(l) - 4 = 0

y( -3) = 0, yCl) ( - 3) = 2, y(^2 ) ( - 3) = 1

e. y" = 2y - 3z'

z" = 3y' - 2z

y(O) = 1, y'(O) = -3, z(O) = -1, z'(O) = 2

f. m1y~ = -k1Y1 + k2(Y2 - Y1)

m2y~ = -k2(Y2 - Y1) - k3Y2

where m1, m2, k1, k2, and k3 are positive constants.

Y1(0) = 0, y~(O) = -2, Y2(0) = 1, y~(O) = 0

g. y' = xy+ z

z" = -x^2 y + z' - 3ex

y(l) = -2, z (l) = 3, z'(l) = 0

12. Specify which initial value problems in exercise 11 are linear and which
    are nonlinear.