155:507 Analytical Methods for Chemical and Biochemical Engineering
Fall 2007: Homework Set 3
Instructor: Prof. Marianthi Ierapetritou
Assigned 09/20 Due: 09/26
1. (3.1:39) (a) Verify that
3
1
xy
and
3
2
xy
are linearly independent solutions of the
differential equation
064
2
yyxyx
on the interval
),(
.
(b) Show that
0
21
)y,y(W
for every real number x. Does this result violates
theorem 3.3? Explain.
(c) Verify that
3
1
xY
and
2
2xY
are also linearly independent solutions of the
differential equation in part (a) on the interval
),(
.
(d) Find a solution of the differential equation satisfying
.)(y,)(y 0000
(e) By superposition principle
2211
ycycy
and
2211
YcYcY
are both solutions
of the differential equation. Determine whether one, both or neither of the linear
combinations is a general solution to the differential equation on the interval
),(
.
2. (3.2:20) (Same as the quiz) The indicated function
)x(y
1
is a solution of the
homogeneous DE. Use the method of reduction of order to determine a second
solution
)x(y 2
of the homogeneous DE and a particular solution of the
nonhomogeneous equation:
x
exyxyyy
)(,34
1
3. (3.3:59) Use a computer as an aid in solving the auxiliary equation of the following
DE:
0032336345153
4
y.y.y.y.
)(
4. (3.3:54) Consider the boundary value problem
.)/(y,)(y,yy 02000
Discuss if it is possible to determine values of λ so that the problem possesses:
(a) trivial solutions and (b) nontrivial solutions.
5. (3.4:41) Solve the following initial value problem where g(x) is discontinuous.
2
0
2
0
20104
x,
x,xsin
)x(gwhere
)(y,)(y),x(gyy
(solve the problem on the two intervals, and then find a solution so that y and
y
are
continuous at
2/x
)
6. (3.5:28) Discuss how the methods of undetermined coefficients and variation of
parameters can be combined to solve the following DE:
x
exxyyy
12
342
