Lecture 3
Suppose
Separation of Variables
-
1st
order
have
we
DE
of
form:
the
* F(y,t)
=
a
differential equ of
f(y).
y
able
is
form
yz
=
ex
=
3
t
-
+
we
in
write
can
form:
the
it
not
separable
t3
would
left
side
4 bc
to
y
+
- y'
a
*
x
+
since
=
x
+
y
+
(x
=
1)
dx
~
Do,
=
+
2x2
en(y+1)
=
x
+
+
1)(y
+
1)
+
+
1
+
e
=
x
+
c
+
c
+
+
+
y
a
separable?
1
+
Stdy ((x
1
+
x(y 1) 1(y 1)
=
separable
Fyl =I
y xy
y
+
is
=
the
DF
xy
=
want
multiplication
we
eeable
F(y)
form
the
->
Problem:ele
over
well, since
ex e
=
DE
of
come
as
=
e- Y.y e
* Any
if
=g(t)
#
y'
this
of fly).
Solving
·
Separable Equations
first example:
a
=
①separate
Goal:
ky
-
② Integrate
-
both sides
enly)
=
only
-
r
Stydy
(witt)
S-rdt
-
kt
C
+
solve,to ekt+
initial
XY
Ackt3 general
=
=
y
value
1
Any=
y
y(t)
solution
a
=
Aet
=
=
solution
diff equ
to
problem:
consider:
Find
if
yF0
valid
=
lyI=Ae
An
differential
equation
such that
initiadition
yz
1
+
#yz
Siyz dy fat
1
=
=
tavi(y)
t
=
2
+
tan(t+c)
y
=
2lution
initial cond:y(0) 1
=
1
an
+
=
(0 C)
+
1 tan (C)
=
c
=
y tan(+
=
y(t)
variables
=
③
find function
i)
+
&
the
IP
see
to
#
Find
explicitsolution
an
=yy(1)
*
Sydy=Sx
=
-
2
dx
Iyz Ex2
=
y2
IVP
the
to
c
+
x2 c
+
=
x
cAb ( ly)a
=
=
y
=
=
y x
c
=
ly1 1
c
=
-
2
c
=
+
-
y
...
3
-
+
3
x
=
=