1.5 Autonomous DE
.
→ indep variable
. doesn't appear explicitly
DX
-
f- (a)
df
examples
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
.
→ indep variable
. doesn't appear explicitly
DX
-
f- (a)
df
examples
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
