Differential Equations                                        Quiz #4                                                 March 18, 2005

Name____________________                   R.  Hammack                                                 Score ______

(1)    The function  y=c_1x+c_2x ln x  is a two-parameter family of solutions of x^2y''-x y' +y=0.

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

2 = c_1 + c_2(1) ln (1)  c_1 = 2

y=2x+c_2x ln x

y'=2+c_2 ln x+c_2

1 = 2+c_2 ln (1)+c_2= 2+c_2


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^2, -x and 1, respectively. Each is continuous and the coefficient x^2of 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)   f_1(x)=x^2+1,     f_2(x)=x+1,       f_3(x)=x^2+2+x     

Notice that  f_1(x)+f_2(x)-f_3(x)=0 so the functions are linearly dependent.

(b)   f_1(x)=e^x,     f_2(x)=e^(x + 2)

Notice that e^2f_1(x)- f_2(x)=e^2e^x-e^(x + 2)e^2=e^(x + 2)-e^(x + 2)=0 so the functions are linearly dependent.