Differential Equations, Dr. Wyels, Spring ‘02
Condensing Section 4.1
DE1: y  2 x 2 y  e x y  sin x , subject to y (1)  2 , y (1)  5 , and y (1)  47 .
A. Existence and Uniqueness
1. Identify n, a n (x) , a n1 ( x) , …, a0 ( x) , and g (x ) for DE1.
2. What conditions does Th’m 4.1 put on a n (x) , a n1 ( x) , …, a0 ( x) , and g (x ) for
DE1 to have a unique solution?
3. Is DE1 an IVP or a BVP? Change whatever is needed to come up with a DE that
is whatever DE1 is not.
B. Linear Independence and Dependence
1. Finish the definition: A set of functions f1 ( x) , f 2 ( x) , …, f n (x) is linearly
dependent on an interval I if
2. Write out how to test the functions f1 ( x)  sin( x) , f 2 ( x)  e 4 x ,
f 3 ( x)  5 sin( x)  13e 4 x for linear dependence,
a) via the definition
b) via the Wronskian
c) Use the back of this page to actually do what you explained in a) and b).
C. Forming General Solutions
1. What is the difference between homogeneous and nonhomogeneous linear DEs?
2. Is it possible that f1 ( x)  sin x , f 2 ( x)  e 4 x , f 3 ( x)  cos x forms a fundamental
set of solutions to some linear 2nd-order DE? Why/ why not?
3. Explain how to form the general solution of a nonhomogeneous linear DE.