18.03 Class 4
, Feb 10, 2010
First order linear equations: integrating factors
[1] First order homogeneous linear equations
[2] Newtonian cooling
[3] Integrating factor (IF)
[4] Particular solution, transient, initial condition
[5] General formula for IF
Definition: A "linear ODE" is one that can be put in the "standard form"
_______________________________



r(t)x' + p(t)x = q(t)

x = x(t)
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r(t), p(t)
are the "coefficients" [I may have called q(t) also a coefficient
also on Monday; this is not correct, fix it if I did.]
The left hand side represents the "system," and the right hand side
arises from an "input signal." A solution
x(t)
is a "system response"
or "output signal."
We can always divide through by r(t), to get an equation of the
Reduced standard form:
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x' + p(t)x = q(t)

x = x(t)
(*)
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The equation is "homogEneous" if
q
is the "null signal,"
q(t) = 0 .
This corresponds to letting the system evolve in isolation:
In the bank example, no deposits and no withdrawals.
In the RC example, the power source is not providing any voltage increase.
The homogeneous linear equation
x' + p(t) x = 0
(*)_h
is separable. Here's the solution,
in general on the left, with an example
(with
p(t) = 2t )
on the right:
x' + p(t)x = 0
x' + 2tx = 0
Separate:
dx/x =  p(t) dt
dx/x =  2t dt
Integrate:
lnx =  int p(t) dt + c
lnx =  t^2 + c
Exponentiate:
x = e^c e^{  int p(t) dt }
x = e^c e^{t^2}
Eliminate the absolute value and reintroduce the lost solution:
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x = C e^{ int p(t) dt}
x = C e^{t^2}
In the example, we chose a particular antiderivative of
k , namely
kt.
That is what I really have in mind to do in general. The constant of
integration is taken care of by the constant
C .
So the general solution to
(*)_h
has the form
C x_h , where
x_h
is
*any* nonzero solution:
x_h = e^{ int p(t) dt}
,
x = C x_h
We will see that the general case can be solved by an algebraic
trick that produces a sequence of two integrations.
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 Spring '09
 vogan
 Linear Equations, Equations, Factors, Derivative, Elementary algebra

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