Tutorial 5: Worksheet
Examples
1. Solve
dy
dt
= ky, k > 0 using separation of variables.
2. Solve
dy
dt
+ 2ty = et−t using an integrating factor.
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3. Write x00 + 4x = 5 as a first order system in matrix notation.
Easy problems
1. Separation of variables. Solve
2. Integrating factor. Solve
dy
dt
dv
dt
= g − kv, with g, k > 0, and v(0) = 0.
+ ty = t.
3. High-order to first-order system. Write mx00 + cx0 + kx = f (t) as first order system.
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Hints for hard problems
1. Rate of change of quantity = rate in − rate out.
2. If
3.
dx
dt
= f (x), then x∗ is a steady-state if
dx dt x=x∗
= f (x∗ ) = 0.
R kt
e cos(ωt) dt = ekt (k cos(ωt)+ωsin(ωt)
+C
k2 +ω 2
Hard problems
1. Hummingbird. According to Klaassen and Lindstrom (1996), Journal of Theoretical Biology, the fuel load carried by a hummingbird, F (t) depends on the rate of intake and the rate
of consumption due to metabolism.
Suppose that the intake takes place at a constant rate α. Consumption increases when the
bird is heavier, e.g. the fuel is consumed at a rate proportional to the amount of fuel being
carried with proportionality constant β.
(a) Write an ordinary differential equation model for F (t).
(b) Find F (t).
(c) Find the steady-state fuel load.
2. Newton’s law of cooling. Newton’s law of cooling states that dT
dt = −k(T − A) where T is
the temperature, t is time, A is the ambient temperature, and k > 0 is a constant. Suppose
that the ambient temperature oscillates: A = A0 cos(ωt).
(a) Find T (t).
(b) What happens as t → ∞?
3. Systems. Write y100 + y1 + y2 = t, y200 + y1 − y2 = t2 as a first order system.
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