Summary WTW256 Cheatsheet

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 =