WTW256: LU 1.5: AUTONOMOUS DIFFERENTIAL EQUATIONS Lecture Notes
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Course
WTW 256 - Differential Equations
Institution
University Of Pretoria (UP)
Lecture notes were made while watching the recorded lectures assigned to watch. These notes include theory (theorems) and worked out examples from the lecturer.
These specific notes cover Autonomous differential equations.
1) DT
gp-=klT-7m# don't see
tin
a small
equation
2) dy = -
kvj
of
3) dy
'
y is function of
'
y2
>e
a
=
zy
-
da
there
implicitly
.
→
•
sometimes sketch BE without solving BE
Wh Example
✗
'
=sihx
to motivate
xlo) =
11
Findxl1)m
ftp.udx-fldtlnlcosecx-cotx/- tnclcosecx-cotx1
solving this
*
=
( et
→ impossible to find explicit solution
?⃝
, Use initial conditions
*
c. cosec 11 cot 11
= -
lcosecx -
cotx / =
( case ex -
cotx )@t
Xli )= ? autonomous get approximation
D. E. for)( (1)
critical points of da of fx=o
=
f- ( x) is the solution CS)
Jp
if x=c is a critical point then xctjyc is a constant
solution of DF ( equilibrium solution)
Facts to keep in mind
CH increasing function
'
✗ > o → x
1.
CH co decreasing function
'
K → a
2.
3. Autonomous DE 's are separable
÷ fact glt)
=
-
4. f
'
continuous →
unique solution C uniqueness theorem)
cberirdtirej
increasing
5. Solutions of autonomous bf → MM @☒• tonic f or
decreasing
-
AND if f ' is continuous , solutions don't
intersect
6. The
graphs of constant solutions =
HORISONTAL lines
is continuous at the critical
'
and iff points then no
other graph can intersect the horizontal lines
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