MATH 2280-002
Lecture Notes: 2/19/2013
Math 2280 Section 002 [SPRING 2013]
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Systems of Differential Equations
Systems of differential equations appear in many contexts. Here are a few examples:
• Coupled Mass-Spring Systems.
Consider a double mass-spring system (with no dashpot).
Two forces act on the mass m1 . By Newton’s second law,
m1 x001 = −k1 x1 + k2 (x2 − x1 ).
One force acts on mass m2 , so
m2 x002 = −k2 (x2 − x1 ).
• Closed Mixing Tank Systems.
Suppose we have a closed system consisting of Tank 1 with volume V1 and Tank 2 with volume
V2 . The concentration of the inflow of Tank 1 is equal to the concentration of the outflow of
Tank 2. Analogously, the concentration of the outflow of Tank 1 is equal to the concentration
of the inflow of Tank 2. Similar statements hold if I replace the word “concentration” with the
word “rate.” I’ll assume both the rates are equal to r. We can model the amount x1 (t) of salt
in Tank 1 and the amount x2 (t) of salt in Tank 2 by the system
x2
x1
−r
V2
V1
x1
x2
0
x2 = r − r .
V1
V2
x01 = r
• Electrical Networks. RLC circuits connected in parallel can be modeled by systems of (linear)
DE’s.
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MATH 2280-002
Lecture Notes: 2/19/2013
The following example will lead us to an important observation.
Example. Write the system of second-order DE’s
m1 x001 = −k1 x1 + k2 (x2 − x1 ).
m2 x002 = −k2 (x2 − x1 ).
into a system of first-order linear DE’s.
We’re going to introduce two new variables v1 = x01 and v2 = x02 . This allows us to rewrite our
system as follows:
m1 v10 = −k1 x1 + k2 (x2 − x1 ).
m2 v20 = −k2 (x2 − x1 ).
v1 = x01
v2 = x02
We have four DE’s instead of two in our system, but now all equations are first-order.
Okay, that’s not particularly clever, but it’s useful. By using this technique, we can always rewrite a
system of DE’s as a system of first-order DE’s. In Chapter 5, we’ll learn the eigenvalue method for
solving systems of linear first-order DE’s with constant coefficients, but now we know that we can
still solve these systems if we drop the “first-order” requirement.
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