Question
Define and in each, give examples of the following terminologies:
1. Partial Differential Equation (PDE).
2. Linear PDE.
3. Quasi-linear PDE.
4. initial conditions.
5. Boundary conditions.
6. What is the difference between Method of Separation of
Variable and Method of characteristics?
Question
State whether the following are True or False. Motivate
your answer.
(a) xu x + yuy = x2 + y2 is homogeneous.
(b) utt + u xxxx = 0 is fifth order.
(c) ut = c2 u xx is an ODE.
(d) y′′ − 4y′ + 5y = 0 is a PDE.
(e) Method of characteristics or Lagranges method. This
method consists of transforming the ODEs to a system of
PDEs which can be solved and the found solution is transformed into a solution for the original ODE.
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Question
−u
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1. Solve for x > 0, xu x + uuy = xe , with u x, x = x.
2. Find the unique solution to au x + buy = 0, where a
and b are constants with a ̸= 0, with the initial condition
u( x, 0) = g( x ).
3. Use method of characteristics to find the general solution
of the equation 5u x + 2uy = 0, with u(0, y) = sin(y).
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4. Find the solution to ut − 3u x = 0, with u( x, 0) = e−x .
5. Solve u x + uy + u = e x+2y with u( x, 0) = 0.
6. Use method of characteristics to solve the Cauchy problem, u x + uy = 1, with u( x, 0) = f ( x ).
Question
1. Show that the linear change of variables
w = x−y
v = x + y,
transforms the equation u xy = 0 into the wave equation
uvv − uww = 0.
2. Which of the following PDEs are linear? Which of those
are homogeneous?
x2 u x + y2 uy = sin( xy),
e x u x + eu uy = 0,
xu xx + yuyy = u x .
3. Find all solutions u = u( x, y) of the equation u xy = xy.
4. Find all separable solutions u = F ( x ) G (y) of the equation
2
xuy = yu x .
5. Find all separable solutions u = F ( x ) G (y) H (z) of the
equation u x − uy + uz = 0.
6. Find all solutions u = u( x, t) of the equation ut + 2xtu x =
et .
7. Find a function f ( x ) for which the initial value problem
u x + uy = 2xu,
u( x, x ) = f ( x )
has no solutions and a function f ( x ) for which it has infinitely many solutions.
8. Show that the characteristic curves for the equation yu x −
xuy = 0 are circles around the origin. Conclude that u( x, 0) =
f ( x ) must be even for any solution u.
9. Find all solutions u = u( x, y) of the equation u x + uy +
u = ey− x .
10. Find all solutions u = u( x, y, z) of the initial value problem
xu x + 2yuy + uz = 3u,
u( x, y, 0) = f ( x, y).
11. Which of the following second-order equations are hyperbolic? elliptic? parabolic?
u xx − 2u xy + uyy = 0,
3u xx + u xy + uyy = 0,
u xx − 5u xy − uyy = 0.
12. For which values of a is the equation au xx + au xy + uyy =
0 elliptic?
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1
Question
1. When is a second order PDE said to be parabolic, elliptic
and Hyperbolic?
2. Consider all the practice problems of Chapter 7 in MTM360S
Course Note.
3. Consider all the practice problems of Chapter 8 in MTM360S
Course Note.
4. Consider all the practice problems 9.1 to 9.10 (pg 74) in
MTM360S Course Note.
5. Consider all the practice problems 10.1 to 10.7 (pg 83-85)
in MTM360S Course Note.
6. Consider all the practice problems 11.1 to 11.15 (pg 90-92)
in MTM360S Course Note.
Keep practising more.
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