_________________________________________________________________________________
Differential Equations                                        Quiz #4                                                 March 18, 2005

Name____________________                   R.  Hammack                                                 Score ______
_________________________________________________________________________________

(1)    The function  y=x+x ln x  is a two-parameter family of solutions of y''-x y' +y=0.

(a)  Find a member of the family satisfying y(1)=2 and y'(1)=1.

y=2x+x ln x

y'=2+ ln x+

1 = 2+ ln (1)+= 2+

=-1

The solution to the I.V.P. is   y=2x-x ln x

(b)  Is your solution from part (a) above a unique solution of the initial value problem  y(1)=2, y'(1)=1? Explain.

Yes. The coefficients of y'', y' and y are , -x and 1, respectively. Each is continuous and the coefficient of y'' is nonzero on an interval containing 1. By theorem 4.1, the solution is unique.

(1)   Decide if the following sets of functios are linearly indepenednt or dependent.

(a)   (x)=+1,     (x)=x+1,       (x)=+2+x

Notice that  (x)+(x)-(x)=0 so the functions are linearly dependent.

(b)   (x)=,     (x)=

Notice that (x)- (x)=-=-=0 so the functions are linearly dependent.