Linear First Order Differential Equations
These equations take the general form
dy
dx
=

p
(
x
)
y
+
g
(
x
)
,
i
.
e
.,
dy
dx
+
p
(
x
)
y
=
g
(
x
)
,
where
p
(
x
) and
g
(
x
) are continuous (piecewise continuous is OK) func
tions on some interval
a
0
≤
x
≤
b
0
. We refer to these first order differ
ential equations as
linear
differential equations because the unknown
y
appears linearly, i.e., to the first power, with a known coefficient,
p
(
x
).
The equation is
homogeneous
if
g
(
x
)
≡
0, for given functions
g
(
x
) other than zero it is said to be
inhomogeneous
.
Homogeneous Case:
Equation:
dy
dx
+
p
(
x
)
y
= 0.
Properties:
•
y
(
x
)
≡
0 is a solution; the
trivial
solution. (Clear.)
•
If
y
(
x
) is any solution of the homogeneous equation then, for any
constant
c
, the multiple
c y
(
x
) is also a solution of that equation. (Just
substitute
c y
(
x
) into the equation.)
•
If a solution
y
(
x
) is nonzero for some value, say
x
1
, of
x
, then it is
nonzero for all values of
x
. (Will be shown.)
Method of Solution:
In the homogenous case we have
dy
dx
=

p
(
x
)
y
. Assuming that we are
looking for a nonzero solution, we can write
1
y
(
x
)
dy
dx
=

p
(
x
)
.
Integrating both sides with respect to
x
, we have (natural logarithm)
log

y
(
x
)

=

Z
x
p
(
s
)
ds
+ ˆ
c
≡ 
P
(
x
) + ˆ
c,
1
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where ˆ
c
is an arbitrary constant and
P
(
x
) is an antiderivative, or in
definite integral, of
p
(
x
). Taking the exponential of both sides we have
the general solution
y
(
x, c
) =
e
ˆ
c
exp
{
Z
x
p
(
s
)
ds
} ≡
c e

P
(
x
)
,
where
P
(
x
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 Spring '10
 ROBINSON
 Equations, lim, Constant of integration, Boundary value problem

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