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4. In Exercises i) and ii) determine whether the given graph has a Hamilton circuit. If it does, find such a circuit. If it does not, give an argument
5. Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an edge in common.
4. In Exercises i) and ii) determine whether the given graph has a Hamilton circuit. If it does, find such a circuit. If it does not, give an argument to show why no such circuit exists.
i)
ii)
3. Proof that an undirected graph has an even number of vertices of odd degree.
2. Describe at least one way to generate all the partitions of a positive integer n. (You can get idea from Exercise 49 in Section 5.3.)
1. Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) For each new numbering plan you discuss, show how to find the number of different telephone numbers it supports.
(i) State the Existence and Uniqueness theorem for the
differential equation of the first order.
(ii) A home buyer can spend no more than $700 per month on
mortgage payments. Suppose that the interest rate is
7% and that the term of the mortgage is 30 years.
Assume that the interest is compounded continuously
and that payments are also made continuously.
i. Determine the maximum amount that this buyer can
borrow.
ii. Determine the total interest paid during the term
of the mortgage
Verify that the function 𝑦 = 𝑐1𝑒
(−𝑘+2𝑖)𝑥 + 𝑐2𝑒
(−𝑘−2𝑖)𝑥 is a
solution to
𝑦
′′ + 2𝑘𝑦
′ + (𝑘
2 + 4)𝑦 = 0
Solve the following boundary value problems.
(i) 𝑦
′′ + 4𝑦 = 0; 𝑦(0) = 3, 𝑦(𝜋/2) = −3,
(ii) 𝑦
′′ − 25𝑦 = 0; 𝑦(−2) = 𝑦(2) = cosh 10.
(iii) 𝑦
′′ + 2𝑦
′ + 2𝑦 = 0; 𝑦(0) = 1, 𝑦(𝜋/2) = 0.
With the use of reduction of order for differential
equations, reduce the following to first order and
solve.
(i) 𝑦
′′ + 𝑒
𝑦𝑦
′3 = 0,
(ii) 𝑥𝑦
′′ + 2𝑦
′ + 𝑥𝑦 = 0, 𝑦1 =
sin 𝑥
𝑥
,
(iii) (1 − 𝑥
2
)𝑦
′′ − 2𝑥𝑦
′ + 2𝑦 = 0, 𝑦1 = 𝑥,
(iv) 4𝑥
2𝑦
′′ − 3𝑦 = 0, 𝑦(1) = 3, 𝑦
′
(1) = 2.5
With the use of reduction of order for differential equations, reduce the following to first order and solve. (i) 𝑦 ′′ + 𝑒 𝑦𝑦 ′3 = 0, (ii) 𝑥𝑦 ′′ + 2𝑦 ′ + 𝑥𝑦 = 0, 𝑦1 = sin 𝑥 𝑥 , (iii) (1 − 𝑥 2 )𝑦 ′′ − 2𝑥𝑦 ′ + 2𝑦 = 0, 𝑦1 = 𝑥, (iv) 4𝑥 2𝑦 ′′ − 3𝑦 = 0, 𝑦(1) = 3, 𝑦 ′ (1) = 2.5.
