MA 2327
Assignment 2
Due 19 October 2015
Id:
2327-f2015-2.m4,v 1.2 2015/10/17 15:16:35 john Exp john
1. Find invariants for the following equations
(a)
3tx2 + t3
dx
=− 3
dt
x + 3t2 x
(b)
dx
3t2 − 1
=
dt
2x
(c)
dx
cos x cos t
=
dt
sin x sin t
2. Show that p and q defined on the set (t, x) 6= (0, 0) by
p(t, x) =
t2
x
,
+ x2
q(t, x) =
t2
t
+ x2
satisfy
∂p ∂q
+
=0
∂x ∂t
but there is no function U defined on the set (t, x) 6= (0, 0) with
∂U
= p,
∂t
∂U
= −q.
∂x
Hint: What are the values of U on the curve t = cos θ, x = sin θ?
1
Id:
2327-f2015-2.m4,v 1.2 2015/10/17 15:16:35 john Exp john 2
3. The equation
dx
= x + et
dt
can be written as
dx
p(t, x)
=
dt
q(t, x)
in a variety of ways. We could, for example, choose
p(t, x) = x + et ,
q(t, x) = 1
or we could choose
p(t, x) = xe−t + 1,
q(t, x) = e−t .
The two are of course equivalent, but show that the first is not integrable, while the second is.
Note: A factor which when multiplied by the numerator and denominator of an equation makes it integrable is called an integrating factor.
Except in special cases there is unfortunately no algorithm for finding
them.