Math 54 ODE review
1
1. Separable Equations
•
When to use:
When you can write it as
y
0
=
f
(
y
)
g
(
x
).
•
How to solve:
Rewrite as
dy
f
(
y
)
=
g
(
x
)dx and integrate both sides.
•
Reminders, Tips, and Tricks:
–
Separable things are good, so always check this first.
If there are things like
y
2
floating
around, it’s a pretty good hint its probably separable (since it’s not linear).
–
Don’t forget about the constant solutions you get from solving
f
(
y
) = 0
–
Remember your +C’s!
2. Linear (FirstOrder) DiffEqs
•
When to use:
When it’s not separable and
y
00
doesn’t appear anywhere.
•
How to solve:
Rewrite equation in the form
y
0
+
P
(
x
)
y
=
Q
(
x
) and compute
I
=
e
R
P
(
x
)
dx
.
Then
y
=
R
IQdx
I
•
Reminders, Tips, and Tricks:
–
Linear means that
y
and
y
0
are never squared, cubed, squarerooted, etc.
–
Remember your +C’s!
3. Linear SecondOrder Homogeneous DiffEqs
•
When to use:
Whenever
y
00
shows up.
•
How to solve:
Write as
ay
00
+
by
0
+
cy
= 0 (
a, b, c
must be constants) and solve the equation
ar
2
+
br
+
c
= 0.
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 Spring '08
 Chorin
 Differential Equations, Linear Algebra, Algebra, Equations, yp, complex roots, yp cos

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