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Question 1
a) Let ᡷ(ᡶ) be the solution of the initial value problem 2ᡶ − ᡷ
ᡖᡶ = 0,^ ᡷ ≥ 0,^ ᡷ(0) = 2
Find the value of ᡷ(4).
b) A hall of volume 1000 ᡕᡥ⡱^ contains air with 0.001% of carbon monoxide. At time ᡲ = 0 the ventilation system starts blowing in air which contains 2% of carbon monoxide by volume. If the ventilation system blows in and extracts air at a rate of 0.3 ᡕᡥ⡱ᡥᡡᡦ⡹⡩ , how long will it take for the air in the hall to contain 0.025% of carbon monoxide? Give your answer in minutes correct to 2 decimal places.
a) ᡷ ᡖᡷ = 2ᡶ ᡖᡶ
1 2 ᡷ
ᡷ⡰^ = 2ᡶ⡰^ + 4
b) Let ᡶ ᡥ⡱^ = volume of CO at time t. ᡖᡶ ᡖᡲ = 0.3 ∙ 0.02 − 0.3 ∙^
20 − ᡶ = 0.0003 ᡖᡲ^ ⇒^ − ln|20 − ᡶ| = 0.0003ᡲ + ᡕ 20 − ᡶ = ᠧᡗ⡹⡨.⡨⡨⡨⡱ぇ ᡲ = 0, ᡶ = 0.01 ⇒ ᠧ = 19. ᡶ = 20 − 19.99ᡗ⡹⡨.⡨⡨⡨⡱ぇ ᡶ = 0.25 ⇒ 19.75 = 19.99ᡗ⡹⡨.⡨⡨⡨⡱ぇ
∴ ᡲ = ln 㐶
Question 2
a) At 11am, a cup of coffee at 86° C is placed in an air-cond room which has a temperature of 23° C. The air-cond is immediately switched off and the room warms up at a uniform rate to 32° C at 12 noon. Assume that at any time between 11am and 12 noon, the rate of the coffee satisfies the equation: ᡖᡆ ᡖᡲ = −(ᡆ − ᡆ〲ぁぉ) where ᡆ〲ぁぉ is the room temperature at that time and the unit of time is measured in hours, find the temperature of the coffee at 11.30am.
b) Solve the differential equation ᡷ䖓䖓^ − ᡷ = 2ᡗけ^ with the initial conditions ᡷ( 0 = 2 , ᡷ䖓( 0 )^ = 1_._
a) ᡆ〲ぁぉ gets from 23° C to 32° C uniformly in 1 hour.
ᡆ〲ぁぉ = 23 + 9ᡲ ᡖᡆ ᡖᡲ = −(ᡆ − 23 − 9ᡲ)
ᡗᔖ 〱ぇ^ = ᡗぇ
ᡆ = ᡗ⡹ぇ^ 㔅 ᡗぇ(23 + 9ᡲ) ᡖᡲ = ᡗ⡹ぇ^ 㐶23ᡗぇ^ + 㔅 9ᡲᡗぇ^ ᡖᡲ㑀 = ᡗ⡹ぇ(23ᡗぇ^ + 9ᡲᡗぇ^ − 9ᡗぇ^ + ᡕ)
b) ’⡰^ − 1 = 0 ⇒ ’ = ±
ᡷ = (ᠧᡶ + ᠨ)ᡗけ ᡷ䖓^ = ᠧᡗけ^ + (ᠧᡶ + ᠨ)ᡗけ ᡷ䖓䖓^ = 2ᠧᡗけ^ + (ᠧᡶ + ᠨ)ᡗけ 2ᠧᡗけ^ = ᡗけ^ ⇒ ᠧ = 1 ᡷ = ᠩᡗけ^ + ᠰᡗ⡹け^ + ᡶᡗけ ᡷ䖓^ = ᠩᡗけ^ − ᠰᡗ⡹け^ + ᡗけ^ + ᡶᡗけ ᡷ(0) = 2, ᡷ䖓(0) = 1 ⇒ ᠩ = ᠰ = 1
∴ ᡷ = ᡗけ^ + ᡗ⡹け^ + ᡶᡗけ
Question 4
a) Find and classify all the equilibrium solutions of the differential equation ᡖᡶ ᡖᡲ = ᡶ
b) Compute the inverse Laplace transform of ᠲ(ᡱ) =
ᡱ⡰^ − 4 ᡱ + 13.
a) 〱け〱ぇ = (ᡶ − 1)⡰(ᡶ + 1)(ᡶ − 2)
b) ᠸ⡹⡩^ 䙲 (^) うㄘ⡹⡲う⡸⡩⡱う⡹⡰ 䙳 = ᠸ⡹⡩^ 䙲 (^) (う⡹⡰)う⡹⡰ㄘ⡸⡱ㄘ䙳 = ᡗ⡰ぇ^ cos 3ᡲ
Question 5
a) In an RLC circuit of inductance 1H, resistance 3 Ω , capacitance ⡩⡰ F and
voltage V volts, it is known that the charge of the capacitor Q satisfies the equation ᡖ⡰ᡃ ᡖᡲ⡰^ + 3
At time ᡲ = 0 s, you observe that there is no voltage applied to the circuit and that both ᡃ = 〱〘〱ぇ = 0 at that time. At time ᡲ = 2 s, you apply a constant voltage of 2V to the circuit and then you switch off the voltage at time ᡲ = 4ᡱ. Find the value of Q at time ᡲ = 5ᡱ.
b) Morphine in the blood decomposes with a half life of 2.9 hours. Suppose a doctor injects 60mg of morphine into a patient and does it again 6 hours later. Find the amount of morphine in mg in the patient’s blood at the seventh hour. You may assume that there is no morphine in the patient’s blood prior to the first injection.
a) 〱
ㄘ〘 〱ぇㄘ^ + 3^
〱〘 〱ぇ + 2ᡃ = 2䙰ᡳ(ᡲ − 2) − ᡳ(ᡲ − 4)䙱, ᡃ(0) =^
〱〘 〱ぇ (0) = 0
ᠸ(ᡃ)䙰ᡱ⡰^ + 3ᡱ + 2䙱 =
(ᡗ⡹⡰う^ − ᡗ⡹⡲う)
⡹⡰う − ᡗ⡹⡲う) = 㐶^1
ᡱ −^
ᡱ + 1 +^
⡹⡰う − 㐶^1
ᡱ −^
ᡱ + 1 +^
⡹⡲う
ᡃ = (1 − 2ᡗ⡹ぇ⡸⡰^ + ᡗ⡹⡰ぇ⡸⡲)ᡳ(ᡲ − 2) − (1 − 2ᡗ⡹ぇ⡸⡲^ + ᡗ⡹⡰ぇ⡸⡶)ᡳ(ᡲ − 4)
∴ ᡃ(5) = −2ᡗ⡹⡱^ + ᡗ⡹⡴^ + 2ᡗ⡹⡩^ − ᡗ⡹⡰^ ≈ 0.
b) 〱〔〱ぇ = −ᡣᠹ, half-life=2.9 hours
ᡣ =
ln 2
ᡖᠹ ᡖᡲ = −ᡣᠹ + 60‒(ᡲ) + 60‒(ᡲ − 6),^ ᠹ(0) = 0 ᡱᠸ(ᠹ) = −ᡣᠸ(ᠹ) + 60 + 60ᡗ⡹⡴う
ᠸ(ᠹ) =
ᡱ + ᡣ +^
⡹⡴う
ᠹ = 60ᡗ⡹〸ぇ^ + 60ᡗ⡹〸(ぇ⡹⡴)ᡳ(ᡲ − 6)
⡵ ⤢⤤ ⡰ ⡰.⡷ (^) + 60ᡗ⡹
⤢⤤ ⡰ ⡰.⡷ (^) ≈ 58.
Question 6
a) Where will the point (1,2,3) move to if we rotate 90° around the y-axis according to the right-hand rule and then rotate 90° around the z-axis according to the right-hand rule?
b) A country has 2 main sectors in its economy: Agriculture (A) and Manufacturing (M). The demand for A and M are $120 million per year and $100 million per year respectively. It costs 40 cents of A to generate $1 of A. It costs 30 cents of M to generate $1 of M. It costs 30 cents of A to generate $1 of M. It also costs 60 cents of M to generate $1 of M. Formulate this as a Leontief input-output model and find the production in millions of dollars per year for A and M.
a) Rotate 90° around y-axis,
㐷
Rotate 90° around z-axis,
b) 䙲ᡄᠲ䙳 〸⡸⡩
⡩ ⡰ −^
⡩ ⡲ 0 ⡩⡲
〸
ぁ
⡩ ⡰ −^
⡩ ⡲ 0 ⡩⡲
ぁ 䙲ᡄᠲ䙳 ⡨
2 − ’^ −
4 − ’㑀 = 0^ ⇒^ ’ =
2 ⇒^ 0ᡶ −
4 ᡷ = 0^ ⇒^ ᡗᡡᡙᡗᡦᡴᡗᡕᡲᡧᡰ 䙲
4 ⇒^
4 ᡷ = 0^ ⇒^ ᡗᡡᡙᡗᡦᡴᡗᡕᡲᡧᡰ 䙲
䙲^10 11 䙳
⡹⡩ = 䙲^10 −1 1 䙳
ぁ
= 䙲^10 11 䙳 㐸
2 ぁ^0
㑂 䙲^10 −1 1 䙳
ぁ
= 䙲^10 11 䙳 㐸
2 ぁ^0
㑂 䙲^10 −1 1 䙳 䙲^81921024 䙳 = 㐸
2 ぁ^ +
4 ぁ^ = 4^ ⇒^ ᡦ = 4
2 ⡲^ +
4 ⡲^ = 452
Question 8
a) Classify the linear systems of ordinary differential equations with matrices (ᡡ) 䙲^11 −2^3 䙳 , (ᡡᡡ) 䙲^28 −2 1 䙳 , (ᡡᡡᡡ) 䙲^21 22 䙳 , (ᡡᡴ) 䙲^08 −0.1−2䙳 , (ᡴ) 䙲^05 −5 0 䙳.
b) Humans and Klingons both live on Planet Zorg, but neither likes the other and both have a tendency to leave Zorg if they consider that the other side is too numerous. Despite this problem on Zorg, 1 million Humans and 2 million Klingons continue to move there from other plates as long as there are both Human and Klingon populations on Zorg. Suppose that we model this situation using the system of ordinary differential equations ᡖᠴ ᡖᡲ = 5ᠴ − 4ᠷ + 1,^
where H and K denote the number in millions of Humans and Klingons respectively at any time t and time is measured in years. If at time ᡲ = 0 , there are 500 million Humans and 〸⡴ of Klingons living on Zorg, what is the minimum value that k must exceed in order that all Humans will be driven out of Zorg after some time?
a) (i) ᡆᡰ = −1, ᡖᡗᡲ = −5 ⇒ ᡅᡓᡖᡖᡤᡗ
(ii) ᡆᡰ = 3, ᡖᡗᡲ = 18, ᡆᡰ⡰^ − 4ᡖᡗᡲ < 0 ⇒ ᡅᡨᡡᡰᡓᡤ ᡅᡧᡳᡰᡕᡗ (iii) ᡆᡰ = 4, ᡖᡗᡲ = 2, ᡆᡰ⡰^ − 4ᡖᡗᡲ > 0 ⇒ ᡀᡧᡖᡓᡤ ᡅᡧᡳᡰᡕᡗ (iv) ᡆᡰ = −0.1, ᡖᡗᡲ = 16, ᡆᡰ⡰^ − 4ᡖᡗᡲ < 0 ⇒ ᡅᡨᡡᡰᡓᡤ ᡅᡡᡦᡣ (v) ᡆᡰ = 0, ᡖᡗᡲ = 25, ᡆᡰ⡰^ − 4ᡖᡗᡲ < 0 ⇒ ᠩᡗᡦᡲᡰᡗ
b) Let ᡰጘ = 䙲ᠴᠷ䙳 , ᠨ = 䙲 −1^5 −4 2 䙳 , ᠲጘ = 䙲^12 䙳, then
ᡖᡰጘ ᡖᡲ = ᠨᡰጘ + ᠲ
ᡆᡰ ᠨ = 7, det ᠨ = 6, ᡆᡰ⡰^ − 4ᡖᡗᡲ > 0 ⇒ ᡀᡧᡖᡓᡤ ᡅᡧᡳᡰᡕᡗ
䚘5 − ’−1 2 − ’−4 䚘 = 0, ’⡰^ − 7’ + 6 = 0 ⇒ ’ = 1,
’ = 1 ⇒ 4ᡶ − 4ᡷ = 0 ⇒ ᡗᡡᡙᡗᡦᡴᡗᡕᡲᡧᡰ 䙲^11 䙳
’ = 6 ⇒ −ᡶ − 4ᡷ = 0 ⇒ ᡗᡡᡙᡗᡦᡴᡗᡕᡲᡧᡰ 䙲−4 1 䙳
