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SOLVING FIRST ORDER DIFFERENTIAL EQUATIONS
Methods for solving Linear, Exact, Separable, Homogeneous, and Bernoulli types
Linear
1. Rewrite in the form
dy
dx
x y
x
+
=
(
)
(
)
terms of
terms of
2. Determine the integrating factor
e
P x dx
(
)
∫
, where
P
(
x
)
is the factor multiplied by
y
above.
3. Multiply the equation by the integrating factor.
4. Discard the center term of the expression.
(This
happens by integrating and taking the derivative.)
5. Rewrite the left term of the equation in the form
d
dx
x y
(
)
new terms of
.
6. Integrate the equation with respect to
x
.
Do this by
removing the
d
dx
from the left side and adding the
dx
∫
notation to the right.
There will be a constant
of integration in the result.
7. Solve for
y x
( )
.
Exact
1. Rewrite in the form
(
)
(
)
terms of
and
terms of
and
x
y dx
x
y dy
+
=
0
that is:
Mdx
Ndy
+
=
0
.
2. The equation is
exact
if the partial derivative of
M
with respect to
y
equals the partial derivative of
N
with
respect to
x
;
¶
¶
¶
¶
M
y
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This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.
 Spring '08
 comech
 Differential Equations, Equations, Bernoulli

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