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Unformatted text preview: MA 36600 MIDTERM #1 REVIEW 3 §2.3: Modeling with First Order Equations.
• Mixing Problems: Say that we have a tank containing a volume V gal of a certain liquid. A
concentration c lbs/gal of a certain substance ﬂows in at a constant rate r gal/min, and a faucet is
opened at the bottom of the tank so that the total amount of liquid in the tank remains constant.
Let Q = Q(t) denote the amount in lbs of this substance at time t, where Q(0) = Q0 .
Concentration of Substance
c lbs/gal
Q(t)/V lbs/gal Flowing In
Flowing Out Rate of Flow
r gal/min
r gal/min The rate of change of the quantity of the substance in the solution is the diﬀerence of the rate which
ﬂows in and the rate which ﬂows out. This gives the following initial value problem:
dQ
r
+ Q = c r,
Q(0) = Q0 .
dt
V
• Interest Rate Problems: Say that we have an account where money accrues interest compounded
continuously at an annual rate r, and that deposits are also made at a constant rate k per year. Let
S = S (t) denote the amount of money in the account at t years, where S (0) = S0 . This gives the
following initial value problem:
dS
− r S = k,
dt S (0) = S0 . §2.4: Diﬀerences Between Linear and Nonlinear Equations.
• Consider the linear initial value problem dy
+ p(t) y = g (t),
y (t0 ) = y0 .
dt
Say that there exists an interval I = t ∈ R α < t < β such that
i. p(t) is continuous on I ,
ii. g (t) is continuous on I , and
iii. t0 ∈ I .
Then there exists a unique solution y = y (t) to the initial value problem. This is the Existence and
Uniqueness Theorem for linear equations.
• Consider the (nonlinear) initial value problem
dy
= G(t, y ),
y (t0 ) = y0 .
dt
Say that there exists a rectangle R = (t, y ) ∈ R2 α < t < β , γ < t < δ such that
i. G(t, y ) is continuous on R,
∂G
ii.
(t, y ) is continuous on R, and
∂y
iii. (t0 , y0 ) ∈ R.
Then there exists a unique solution y = y (t) to the initial value problem. This is the Existence and
Uniqueness Theorem in complete generality for ﬁrst order equations.
• By determining the regions of R ⊆ R2 where both G and ∂ G/∂ y are continuous, one can determine
the largest rectangle in which the initial value problem has a unique solution. §2.5: Autonomous Equations and Population Dynamics.
• Consider an ordinary diﬀerential equation in the form y = G(t, y ). We say that it is autonomous if
the variable t does not explicitly appear in the equation i.e.,
G(t, y ) = f (y ).
• Say that y = y (t) is a solution to y = f (y ). Then the “limiting value”
yL = lim y (t)
t→∞ ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.
 Spring '09
 EdrayGoins
 Equations

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