Differential Equations Review Sheet
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4Differential Equations Review Sheet 1, Fall 2017
I. First Order DEs
(a) Separable
dy
Form:
= f (x)g(y)
dx
dy
To solve: arrange it like so:
= f (x) dx, integrate both sides!
g(y)
Don’t forget the
lost solutions
y = c, where g(c) = 0.
(b) Linear
General Form: a(x)y + b(x)y = c(x)
Standard
form: y + p(x)y = f (x)
To solve:
Put in standard form (by dividing by a(x) if necessary).
p(x)dx
Compute the homogeneous solution: y
(x) = e
.
h
q(x)
Use the variation of parameters formula:
y(x) = y
(x)(
dx + C).
h
y (x)
x
q(u)
There is also the definite integral solution: y(x) = y
(x)(
du + C),
h
a
y (u)
where C = y(a)/y
(a).
h
II. Linear Constant Coefficient Homogeneous DEs
(a) Linear, Constant Coefficient
1. What does it look like?
Well, it’s linear, and has constant coefficients. :)
We write it as P (D)x = 0, where P (r) is a polynomial.
2. How do I solve it?
2
Get the
characteristic
polynomial: replace y by 1, y by r, y by r
etc. (This
rt
comes from guessing y = e
as a trial solution.)
3. Solve for the roots of the equation containing r (=
characteristic
equation).
r
t
r
t
4. Take roots, r
, r
etc. and arrange as:
y = c
+ c
+ . . .
e
e
1
2
1
2
1
2
5. If roots are complex in the form of a
bi, and you want a
real valued
solution,
at
at
then make them: y = c
cos(bt) + c
sin(bt) + . . .
e
e
1
2
rt
rt
6. If r is a double root, then e
and te
are both homogeneous solutions.
(b) Damping (for my + by + ky = 0)
2
1.
Underdamping
when b
4mk < 0, so roots are complex, solutions oscillate.
2
2.
Overdamping
when b
4mk > 0, so roots are real, solutions are exponentials.
2
3.
Critical damping
when b
4mk = 0, so roots are repeated,
bt/2m
bt/2m
solution is y = c
e
+ c
t e
.
1
2
(c) Stability
1. y(t) = 0 is the equilibrium solution.
2. Physics: the system is stable (really asymptotically stable) if the output to the
unforced system always goes to the equilibrium as t
.
3. Math: the system is stable if all charactericstic roots have negative real part.
4. Equivalently: the system is
stable
if all homogeneous solutions to the DE go to
0 as t
.
5. For my + by + ky = 0 the system is stable exactly when m, b and k all have
the same sign (usually positive).
III. Complex Numbers
iθ
(a)
Euler
formula: e
= cos(θ) + i sin(θ).
iθ
2
2
(b)
Polar
form: a + ib = re
, r =
+ b
, tan θ = b/a. (Remember how to draw the
a
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