Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 5
Exercise 5.1
Solve the initial value problem
y 4y + 3y = 0, y (0) = 1, y (0) = 1
Describe the behavior of the solution y (t) as t and t .
Solution.
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 11
Exercise 11.1
Solve
(y u)ux + (u x)uy = x y
1
with the condition u(x, x ) = 0.
Solution.
The characteristic equations are
dy
du
dx
=
=
.
yu
ux
xy
W
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 10
Exercise 10.1
Find the characteristics of the PDE
xux yuy = u.
Solution.
dy
y
We have dx = x . Solving this ODE by the method of separation of vari
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 12
Exercise 12.1
Classify each of the following equation as hyperbolic, parabolic, or elliptic:
(a) Wave propagation: utt = c2 uxx , c > 0.
(b) Heat c
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 13
Exercise 13.1
Show that if v (x, t) and w(x, t) satisfy equation 13.1 then v + w is also a
solution to 13.1, where and are constants.
Solution.
Let
Theory of First Order Linear Ordinary Dierential Equations
with Constant Coecients
Theorem describing the general solution of a rst order linear ordinary differential equation with constant coecients.
Theorem. Suppose that the following conditions hold: (
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 14
Exercise 14.1
Show that if u(x, t) and v (x, t) satisfy equation (14.1) then u + v is also a
solution to (14.1), where and are constants.
Solution.
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 9
Exercise 9.1
Use the coordinate method to nd the solution to ut + 3ux = 0, u(x, 0) =
sin x.
Solution.
Let v = 3x + t and w = x 3t. Then ux = 3uv + u
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 8
Exercise 8.1
Classify each of the following PDE as linear, quasilinear, semi-linear, or nonlinear.
(a) xux + yuy = sin (xy ).
(b) ut + uux = 0
(c) u
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 2
Exercise 2.1
Dene fn : [0, 1] R by fn (x) = xn . Dene f : [0, 1] R by
f (x) =
0 if 0 x < 1
1
if x = 1
(a) Show that the sequence cfw_fn converges p
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 1
Exercise 1.1
Compute the partial derivatives indicated:
(a) x (y 2 sin xy )
2
2
(b) x2 (ex y )
4
(c) xy2 z (z ln x2 /y )
Solution.
(a) x (y 2 sin xy
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 3
Exercise 3.1
Solve the IVP: y + 2ty = t, y (0) = 0
Solution.
Since p(t) = 2t, we nd (t) = e
2
by et to obtain
2tdt
2
= et . Multiplying the given eq
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 4
Exercise 4.1
Solve the (separable) dierential equation
2 ln y 2
y = tet
.
Solution.
At rst, this equation may not appear separable, so we must simpl
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 7
Exercise 7.1
Determine a and b so that u(x, y ) = eax+by is a solution to the equation
uxxxx + uyyyy + 2uxxyy = 0.
Solution.
We have uxxxx = a4 eax+
Arkansas Tech University
MATH 4343: Partial Dierential Equations
Dr. Marcel B. Finan
Solution to Section 6
Exercise 6.1
Classify the following equations as either ODE or PDE.
(a) (y )4 +
(b)
u
x
t2
(y )2 +4
+ y u =
y
=0
y x
y +x
(c) y 4y = 0
Solution.
(a)
Theory of Second Order Linear Ordinary Dierential Equations
with Constant Coecients
Theorem describing the general solution of a second order linear dierential
equation with constant coecients.
Theorem. Suppose that each of the following conditions hold: