Teoretisk Fysik KTH
FYSMAT (SI1143), Homework 11
1. For each of the following problems, answer the following questions: Is there a
unique real-valued solution y(x)? Find the solution for those problems that have
a unique real-valued solution.
(a)
(b)
(c)
(d)
(e)
y 0 (x) + y(x)2 = 0, y(0) = 1, y 0 (0) = 2.
y 00 (x) + 4y(x) = 0, y(0) = 0, y 0 (0) = 1.
y 000 (x) = 1, y(0) = y 0 (0) = 2.
y 000 (x) = 1, y(0) = y 0 (0) = y 00 (0) = 2.
y 00 (x) − 4y(x) = 0, y(0) = 0, y 0 (0) = 1.
2. For each of the following problems, find the function y(x) satisfying all given
conditions.
(a)
(b)
(c)
y 00 (x) + 4y(x) = sin(x), y(0) = 0, y 0 (0) = 1
y 00 (x) − 4y(x) = 1, y(0) = 0, y 0 (0) = 1
y 0 (x) + 2y(x) = x2 , y(0) = 1.
3. A stone is dropped at time t = 0 from a heights 10 m over the ground. At what
time t > 0 does the stone hit the ground? Formulate an appropriate mathematical
model to answer this question, and introduce and explain all mathematical symbols
that you need. You can ignore friction (but, of course, you should take into account
the gravitational force). Note: You probably know the solution of this problem
from high school, but the point is that you are careful about formulating and
solving a mathematical model.
4. Formulate a simple mathematical model of a violin string (that is mounted on
a violin) taking into account the gravitational force and such that your model
has a unique solution. Assume that the string is parallel with the surface of the
earth. Introduce and explain all mathematical symbols that you need. Note: The
solution of this problem is pretty much given in the course book but, again, the
point is that you are careful about formulating and solving a mathematical model.
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Version 1 (v1) by Edwin Langmann on January 25, 2012; revised by EL on January 25, 2012
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