Differential Equations  Final Exam
By Shun-Feng Su
Jan. 13, 2014
一. True/False. Be sure to justify your answers. 5 points each.
(1) If  (x) is an integrating factor of a differential equation, then
k  (x) is also an integrating factor of the differential equation for
any non-zero constant k.
(2) The supposition principle is feasible for any linear differential
equations no matter what order the differential equations have.
(3) Suppose that f(t) is integrable for t >0 and a>0. Then

 f (t ) (t  a)dt  f (a) .
0
(4) Suppose that p(x) , q(x) , and f (x) are continuous on I. Let
(x ) be the solution of y   p( x) y   q( x) y  f ( x) ; y ( x 0 )  0 ,
y ( x 0 )  0 on I. Then  ( x 0 )  0 for all x  I .
(5) The power series expansion of a function about a fixed point is
unique.

(6) If
 an
n 0
converges, then lim a n  0 .
n
(7) If P( x 0 )  0 and Q(x) and R(x) are both analytic at x 0 , then
x 0 is a singular point of P( x) y   Q( x) y   R( x) y  0 .
(8) If P( x) y   Q( x) y   R( x) y  F ( x) have singular points, then it is
impossible to solve it by the power series method.
(9) If the different equation P( x) y   Q( x) y   R( x) y  F ( x) can be
solved by the power series method, then we can always find two
linearly independent power series solutions.
(10) The Frobenius solution of P( x) y   Q( x) y   R( x) y  0 is valid
in the entire analytic region of ( x  x0 )
Q( x )
R( x)
and ( x  x0 )2
.
P( x)
P( x)
二. Short Answers. 5 points each.
(1) Consider an initial value problem
dx
 f ( x, y ) ; x( y 0 )  x 0 . What
dy
are the conditions for the initial value problem has a unique
solution in the neighborhood of ( x 0 , y 0 )?
(2) Let (x ) be the solution of y   p( x ) y   q( x) y  f ( x ) ;
y ( x 0 )  A , y ( x 0 )  0 . Let (x) be the solution of
y   p( x ) y   q( x) y  f ( x ) ; y ( x 0 )  0 , y ( x 0 )  B . Then which
initial value problem has ( x)  ( x) as its solution?
(3) Given any three functions y1 , y 2 , and y 3 , then what is their
Wronskian?
(4) What is the definition for n functions, y1 , y 2 ,  , y n to be linearly
independent?
(1) n1 2n1
( x  2) n3 .

2
2 n 1
n 1 n ( n  2)5

(5) Find the convergent interval for
(6) Find the convergent radius for the Taylor expansion series of
x 1
about x=2.
( x 2  4 x  13)

(7) Rewrite  2(n  r )(n  r  1)cn x
nr
n 1

+  2( n  r )cn x nr 1 +
n 1



n2
n 0
n 0
 (n  r )(n  r  1)c n x nr +  3c n x n  r  cn x nr1 into one
summation.
(8) In which region is the power series solution of
y  p( x) y  q ( x) y  f ( x) valid?
(9) Define all singular points (regular or irregular) for
( x  1)sin( x) y  ( x   ) y  (3x  1)1 y  ln( x  2) .
(10) While using the Frobenius series in solving a differential equation,
why we need to assume that the leading coefficient c 0 should be
not zero?
三. (10 points) Find the Taylor expansion for ln( x 2  2) about x 0  1 .
四. (15 points) Find the first five terms of the power series solution for
y  
1
1
y 
y  2 ; y (0)  y(0)  3 .
x 1
x2
五. (15 points) Find the power series solution in powers of x1 (i.e., x 0 =1)
for y  y  0 . (Other approaches will not be credited.)
六. (15 points) Find the first five terms of the power series solution for
y  y  (1  x  x 2 ) y  5 through the recurrence relation.
七. (15 points) Solve the differential equation 12 x 2 y  5xy  (1  2 x 2 ) y  0
by the method of Frobenius (at least five nonzero terms for series).
八. (20 points)Solve the differential equation xy   (6  x ) y   3 y  0 by the
method of Frobenius (at least five nonzero terms for series).
九. (20 points)Solve xy  3 y  y  0 by the method of Frobenius (at least
five nonzero terms for series).
The total score is 210. Do your best and good luck.
Have a nice vacation and a happy new lunar year.