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Diﬀerential 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

1

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