Class notes
Class notes MATH 2080 (MATH2080)
- Course
- MATH 2080 (MATH2080)
- Institution
- Clemson University
Provides a general overview of solving methods for differential equations with examples. Examples include all work and the answers.
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. 4-motion3 of a falling body
ef e')
+
Gmm
& u = =
- bu
= C
with-
v(0)
endup
+
M =
vo
mg
W
distance traveled :
x(t) X(0) = 0
,
set origin as release height
x( +) = fv()d +
3 Ns/m
example. 3
kg 500m b =
m= Xo =
Vo 0 m/s 9 814/s2
g
= = -
.
3 =
39 -
3v u = e = et x( +) = fg() -
e
-
)d+
+
g(1 et)
"
5
+ v =
g v(t) =
-
x( +) =
g + +
ge +C
v(0) = 0 0 =
g(0 +
e) +
C= g -
C = -
9 81.
500 =
g
+ +
ge -
g
++ -
500
+ ~
from
Hab
+ = 51 .
97
, Chapter 4 -
2nd order ODEs
a y" + by +
cy = 0
·
a, b
, c are constants
O right hand side is
"homogenous and order" system
·
will consider flt) on right as well
linearly independence
definition -
two functions y , (t) + yalt) are linedly
independent if one function is not a
constant multiple of the other
y , ( + ) E Cyz( +) for all +
e . Sinc + cos(t) are linearly independent
ex - eat-eat are linearly independent
to verify-calculate Wranghian
WClyvel] = Y, Ye =
Y , Ye'- yey : O
Y : 'Ye'
* if = O , linearly dep.
ex .
ritra , yalth = et
, yaltlagat are
linearly independant
W2y .. y2] :
e
+
(rie2 )-ea (re")
+ +
(2) +
N2e(
(2) +
e(
+ +
= -
r,
editrel (re
+
= -
1) = 0
u
20
, (a = 0)
Solving ay" + by +
cy = 0
suppose y(t) : e't for some constant
plug y' wert ree'
+
, y"
:
in y =
,
=> 2e + bre
+
"
a - + + ce = 0
Il factor eut
O
neuer :
-e +
(ara + br + c) = 0
2
a + br + c = 0 -
characteristic equation
2) solve quadratic
3 cases of intrest
·
2 real roots (ritra)
I repeated roo+ (vi)
2 complex roots (r Fra) .
Superposition principle
for and order ODE, 2 linearly independent solutions
Y , (t) + ya(t) yield a
general solutions
( +) C,
y
=
y ,
+
(2ye(t) for all ( , ki
·
we can verify this as a solutio
a [(yi + (2y2]" + b(xy +
+
ceyz] + c(( ,
+ (eye]
* a 2(y, + Ce ya)" =
a2Gy ,
"
+ Ceye"]
7 C(ay ,
"
+ by , +
2y , ) + G(ayz" = by + cye)
thus is a solution for in
y any 1111
* some can be done for 3rd order ODE's
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