Homework 3
For the following four problems, determine whether the existence of a unique
solution is guaranteed by Theorem 1.
1.
2.
3.
4.
dy
dx
= 2x2 y2 ; y(1) = 1
dy
dx
= x y; y(2) = 2
dy
dx
= x y; y(2) = 1
y
p
p
dy
dx
= x 1; y(1) = 0
5. Verify that if c is a constant, then the function dened piecewise by
y (x) =
(
0;
forx c
3
(x c) forx > c
satises the dierential equation
y 0 = 3y 2=3
for all x. Can you also use the \left half" of the cubic
y = (x
c)3
(0)
in piecing together a solution curve of the dierential equation (see Figure
1.3.25). Is there a point (a; b) of the xy-plane such that the initial value
problem
y = 3y 2=3 ; y (a) = b
has either no solution or a unique solution that is dened for all x?
0
For the following two problems, nd the general solutions of the given differential equation.
6.
7.
dy
dx
+ 2xy2 = 0
yy 0 = x(y 2 + 1)
1
For the following two problems, nd explicit particular solutions of the given
initial value problems.
8.
9.
dy
(tan x) dx
= y; y 2 = 2
dy
dx
= 2xy2 + 3x2 y2 ; y(1) = 1
10. A certain moon rock was found to contain equal numbers of potassium
and argon atoms. Assume that all of the argon is the result of radioactive
decay of potassium (its half-life is about 1:28 109 years) and that one of
every nine potassium atom disintigrations yields an argon atom. What is
the age of the rock, measured from the time it contained only potassium?
11. A pitcher of buttermilk initially at 25o C is to be cooled by setting it on
the porch, where the temperature is 0o C. Suppose that the temperature
of the buttermilk has dropped to 15o C after 20 minutes. When will it be
at 5o C?
12. Suppose that a uniform exible cable is suspended between two points
(L; H ) at equal heights located symmetrically on either side of the xaxis. Principles of physics can be used to show that the shape of the
hanging cable satises the dierential equation
a
s
dy 2
1
+
;
=
dx2
dx
d2 y
where the constant a = T= is the ratio of the cable's tension at its lowest
point x = 0 (where y'(0) = 0) and its (constant) linear density . If we
substitute v = dy=dx, dv=dx = d2 y=dx2 in this second order dierential
equation, we get the rst order equation
a
dv
dx
p
= 1 + v2 :
Solve this dierential equation for y (x) = v(x) = sinh(x=a). Then integrate to get the shape function
0
y (x) = a cosh
x
a
+C
of the hanging cable. This curve is called a catenary, from the lating word
for chain.
2