MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: EXAMPLE SHEET1 IV
Questions for supervision classes
Hand in the solutions to questions 1a,c,d and 3. Attempt
all other questions too and raise any problems with your
superviso
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: SOLUTIONS TO EXAMPLE SHEET1
I
1. Classifying ODEs
(a) Nonlinear because of the squared u in the second term on the LHS.
Non-autonomous because the independent variable, x, app
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: SOLUTIONS TO EXAMPLE SHEET1
II
1. Existence, uniqueness and graphical solutions
(a) To apply the existence and uniqueness theorem, rewrite the ODE in its standard from y = f (
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: EXAMPLE SHEET1 V
Questions for supervision classes
Hand in the solutions to questions 1a,b,c(i,ii) [or 0a,b,c]
and 3. Discuss any problems with your supervisor. The
remaining
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: SOLUTIONS TO EXAMPLE SHEET1
III
1. Existence and uniqueness for linear second-order ODEs
(a) Rewrite the ODE
x2 y 2 x y + 2 y = 0
in its standard form y + p(x) y + q(x) y = r(
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
MATH10222: SOLUTIONS
1
1
V
0. Mechanics-free substitute for question 1
(a) The homogeneous solution of the ODE is given by
xH (t) = B cos(t) + C sin(t)
for arbitrary constants C and D. [No
1
Existence and uniqueness theorem for 1st order ODEs
Consider the rst-order ODE in its explicit form
dy
= f (x, y),
dx
subject to the initial condition
y(X) = Y,
(1)
(2)
where the constants X and Y are given.
Theorem
(x,y)
If f (x, y) and f y are continu
1
INTRODUCTION
Notation, Denitions and What are the issues?
The Derivative
Given a function
y(x)
where
x is the independent variable,
y is the dependent variable,
the derivative is dened as
y(x + h) y(x)
dy
= lim
y (x) =
h0
dx
h
y(x) y(x h)
= lim
.
h0
h
1
Boundary and initial conditions
An m-th order ODE must be augmented by m constraints
(in the form of boundary or initial conditions) if there
is to be a unique solution.
Notes:
This is a necessary, not a sucient condition: Even if an mth order ODE is a
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
MATH10222: SOLUTIONS
1
1
IV
1. Inhomogeneous linear second-order ODEs with constant coecients
(a) Exploiting linearity
i.
y + 3 y + 2 y = 4 e2 t
(I)
Corresponding homogeneous equation:
y
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: EXAMPLE SHEET1 III
Questions for supervision classes
Hand in the solutions to questions 1, 2a,b and 3a-f. [Feel
free to skip the application of the initial conditions in
quest
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: EXAMPLE SHEET1 II
Questions for supervision classes
Hand in the solutions to questions 1, 2a, 3a, 4b and 5b.
Attempt all other questions too and raise any problems
with your s
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: EXAMPLE SHEET1 I
Questions for supervision classes
Hand in the solutions to all questions on this sheet for
your supervision classes. If you have a new supervisor
and the proc
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: SOLUTIONS TO EXAMPLE SHEET1
0
1. Integration as the inversion of dierentiation
(a) The derivative of
y1 (x) = sin(x)
is (obviously!)
dy1
= cos(x)
dx
and this answer is unique.
MATH10222 http:/www.maths.man.ac.uk/~ mheil/Lectures/FirstYearODEs
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MATH10222: EXAMPLE SHEET1 0
Questions for supervision classes
This example sheet contains a few warm up exercises
for the course. None of the questions are technically
dicult if you thin