NTW 256
DE
Ender
Explicit Sol .
y
= He
Implicit Sol .
F(y) =
G(x) + C
④
①IVP has a
unique
sol.
Uniqueness theorem for
separable DE
:
·
I [a b] x [c
=
, , d]
f(x y) KEF bee
f'(y) J
·
both
·
,
need to is cont .
on .
I
I.
g(x)
·
cont .
on is cont .
on .
bkael be]
y(a)
·
=
,
Separable Equation first ↑
: A order has sol.
a
unique
add ficgc
c) (giydy (f(x) =
dec
A computeforisable .
Natural Growth Newton's of
+
Decay :
Law
Cooling
:
↑ T'(t) = -
k(T -
A)
& constant
temp
x'(t)
.
* =
kx(keR) of environment
.
·
Need initial condition . i . .
e (0) =
C & time = O
·
Need A-constant temp
.
·
Need another condition
. .e
i. x(t) =
Cz
·
Need initial condition ie
. . T(o) = C ,
Need another condition .
salveusingseperable general
·
of
sol.
·
Solve
using separable eq
.
T(t) A A)ek(t to)} General sol.
-
* = +
(T -
Torricellis Law :
area of 0 of container
e
A(h) d KWh'y starting :
Ahwg
* = -
w
heig
of -
rate
of
change
ht hole (Ahah 0)
, heig
=
.
·
Need
Need initial (h(to) ho)
·
condition =
, Linear First-order DE :
③ .
Uniqueness theorem for LFODE :
d P()y Q(x)} Standard form .
: ya) cort.
* + =
of an interval I
·
find U() =
eSPld J integrating factor. ·
a El
"at
"
Remember sol
throughout
·
most
multiply
·
.
one .
(U()y)
·
reduce to :
=
U(x) &(x)
both
Integrate
·
sides
.
*
(u()) x(u(aq(x)dx + C
y
=
mentation
Methods :
④ Linear
Expression : ④ Homogeneous Expression :
dy +y F(i) 3
·
F(ax +
by c)) Standard Forme Standard Form
= + * =
·
Let
by
·
Let v = = =
(x +
-x() (1)
y
=
v = ax + + c
Vx
..
=
Find
·
r
a
Ad 3
+
Note : F(V).
=
=
·
Sub (I) into standard form
.
bF(v
x F()
· = a + ·
V + =
·
Ja du =
(Idx
finder 3 separable
.
+
bec
·
Solve
St
· dr =
eq.
·
Solve .
Note
* Since
separable
:
we We use
use
eq .
the theorem for
uniqueness separable
to deduce solution
.
q
.
e a .
*
At times it be to
might easier use .
(v + Vxx) dx
dy =
DE
Ender
Explicit Sol .
y
= He
Implicit Sol .
F(y) =
G(x) + C
④
①IVP has a
unique
sol.
Uniqueness theorem for
separable DE
:
·
I [a b] x [c
=
, , d]
f(x y) KEF bee
f'(y) J
·
both
·
,
need to is cont .
on .
I
I.
g(x)
·
cont .
on is cont .
on .
bkael be]
y(a)
·
=
,
Separable Equation first ↑
: A order has sol.
a
unique
add ficgc
c) (giydy (f(x) =
dec
A computeforisable .
Natural Growth Newton's of
+
Decay :
Law
Cooling
:
↑ T'(t) = -
k(T -
A)
& constant
temp
x'(t)
.
* =
kx(keR) of environment
.
·
Need initial condition . i . .
e (0) =
C & time = O
·
Need A-constant temp
.
·
Need another condition
. .e
i. x(t) =
Cz
·
Need initial condition ie
. . T(o) = C ,
Need another condition .
salveusingseperable general
·
of
sol.
·
Solve
using separable eq
.
T(t) A A)ek(t to)} General sol.
-
* = +
(T -
Torricellis Law :
area of 0 of container
e
A(h) d KWh'y starting :
Ahwg
* = -
w
heig
of -
rate
of
change
ht hole (Ahah 0)
, heig
=
.
·
Need
Need initial (h(to) ho)
·
condition =
, Linear First-order DE :
③ .
Uniqueness theorem for LFODE :
d P()y Q(x)} Standard form .
: ya) cort.
* + =
of an interval I
·
find U() =
eSPld J integrating factor. ·
a El
"at
"
Remember sol
throughout
·
most
multiply
·
.
one .
(U()y)
·
reduce to :
=
U(x) &(x)
both
Integrate
·
sides
.
*
(u()) x(u(aq(x)dx + C
y
=
mentation
Methods :
④ Linear
Expression : ④ Homogeneous Expression :
dy +y F(i) 3
·
F(ax +
by c)) Standard Forme Standard Form
= + * =
·
Let
by
·
Let v = = =
(x +
-x() (1)
y
=
v = ax + + c
Vx
..
=
Find
·
r
a
Ad 3
+
Note : F(V).
=
=
·
Sub (I) into standard form
.
bF(v
x F()
· = a + ·
V + =
·
Ja du =
(Idx
finder 3 separable
.
+
bec
·
Solve
St
· dr =
eq.
·
Solve .
Note
* Since
separable
:
we We use
use
eq .
the theorem for
uniqueness separable
to deduce solution
.
q
.
e a .
*
At times it be to
might easier use .
(v + Vxx) dx
dy =
