M2AA1 Diferential Equations: Problem Sheet 5
1. Solve the following boundary value problem: y + y = x2 , y (0) = 0, y (/2) = 1.
Answer: The general solution of the dierential equation is 2 + x2 + c1 cos(x) +
c2 sin(x). The boundary conditions give 2 + c1
M2AA1 Diferential Equations: Problem Sheet 9
1. Consider x = (3 y x)x, y = (x 1 y)y. This is a predator-prey model in which
one has species x has one constant source of food, and species y has limited growth.
(a) Draw the regions where (x > 0, y > 0), (x
M2AA1 Diferential Equations: Problem Sheet 8
1. Take x = x + y 2 , y = y + x2 . Compute the power series expansion of the unstable
manifold through 0 up to order 3.
Answer: The unstable manifold goes through 0 and is tangent to the unstable
10
eigenvector
M2AA1 Diferential Equations: Worked Answers Problem Sheet 2
NOTICE THAT ASSIGNMENTS 1-4 ALL USE ALMOST THE SAME ARGUMENTS.
For more details see the Panopto recordings.
1. Consider a dierential equation x = f (x) where f : W Rn (and W is an open
subset of
1
M2AA1 Diferential Equations: Problem Sheet 3
1. Let A =
ab
b a
. Let us compute exp(A) in an alternative way to what was done
in Example 22. Write A = + R where =
a0
0a
and R =
0b
b 0
.
(a) Show that R = R and therefore that exp(A) = exp() exp(R).
Answe
M2AA1 Diferential Equations: Problem Sheet 1
Question 1
It is a straightforward calculation with the quotient rule for derivatives, to show
that the given solution actually satises x = x(1 x). Since x(1 x) is bounded
and Lipschitz on any bounded set U, on
M2AA1 Diferential Equations: Problem Sheet 7 (discussed in class)
1. Find the extremals of the integral I [y ] =
is of the form
1
0 f (x, y, y
)dx when the integral f (x, y, y )
(a) f [y ] := f (x, y, y ) = y 2 (y )2
d
fy [y ] = 2 . fy [] = 2y . So E-L be
1
M2AA1 Diferential Equations: Problem Sheet 4
1. Consider a 2 2 matrix A. Let 1 , 2 be its eigenvalues. Let t (p) the solution of
x = Ax and x(0) = p. (Called the ow). Assume that 1 < 0 < 2 . Show that
W s (0) = cfw_p R2 ; t (p) 0 as t and W u (0) = cfw
M2AA1 Diferential Equations: Problem Sheet 6
1. Take y + qy = 0 with q (x) > 0. Write this in the form
Show that (whenever y = 0) the vector
. (Hint: show that the angle of
y
y
y
y
and
y
y
=
01
q 0
y
y
.
always moves clockwise as t
y
y
=
y
qy
is always
Imperial College London
M2AA1 Diferential Equations: Practise Exam Paper May 2013
This paper contains more questions than the real paper, in order to encourage
you to cover all topics of the material. You should therefore allow yourself to
spend 2 1/4 hou