One Page Summarize Calculus 3 (some
Separable
Erration
parts
Homogeneous
STEPS
1) Check for Hono (Put I in
STEPS
↳*
f(x Y)
=
2) let u =
,
.
g(x)
=
dy
.
g(x)
=
.
Ydx
4) Sub > ecration above
-
ex
similar
~ 5)
steps
stays with
stays with
I
X
X
=
because
N
F
..
X
(Mdx
=
F(X Y)
,
it's dx already
2xy
=
= 24/
so
u +
2y
cos d.
+
=
cosy
+
24V
EXACT !
or
F
constant
constant is
is
x
-
ANS1
=
orF(X Y)
g(y)
+
of
u(X)
,
/
=
Ans
+
or
N
7) sub back to get func. of
M=
solve for these constant by
S
Y(X) .
g(x)
=
by
can see
M
dy
the original ear. is by
just integrate !
Now
SNdy
=
dx
get func.
6) Integrate
(g(x)dx
u +
xy
M= y
Eroim
:**
du
+
EVEXACT !
This is when
Equation
.
-
h(y)dx
.
EX
x
3) Try to transform f(x, >= g( * )
-
Snedy
u
=
+
h(y)dx
d =,
=
=
uX
=
factor it out ->tf(x)
then
n dis
↓ factor it !
dx
y
+
Diff
EXACT
Equation
~
Diff
Non-EXACT
Erration
.
3) use the erration that works
1)
4) Do
this ! M(Xory)
2) let's try
"Gas My "Tip ! glass
My
-
:
L o #2
=
integrating
Basically
factor
the
equation
Homogeneous
as
same
,
↳ The result
are
EX
EXACT
now
do the same
. xy
+
0
(n(y)
xT
In (4)
=
c
+
=
-
-
-
=
the func of ONLY "X"
(Ide -xdx
fees ? - to use this equation it needs to be
=
M
the func of ONLY "y
-
141
c
+
2
=
I
"
-
etH
=
+ C
F
Ans
+
Y
.
Nx-me
-
=
exact equ .
as
f(x) ? > to use this equation it needs to be
packs
:
erration
.
f(x 4)
·
now
N
eSde Y
=
Diff
linear
These are
u(y) = e
to make it exact !
↳ Sol #1
M(X)
or
NON-EXACT
·
dX
in this erration
it order
value
sub the
now
+c
e
=
.
1) Try to make
d
2)u(x)
M(X)
P(xy
,
TIP !
(4 M(x)
(P(X)dX
fd(y u(x)
EX
.
Y'-Xy
.
y
Differential Equation
Rewrite the
The form
y'
:
=
f(x y
,
f(X 4)
=
=
.
Xn 1
,
+
=
=
X
;
X
+
xy
=
X
+ x
Xn +
h
=
I
f f(x)dx
0
.
5
from
Xo
hYn
+
=
STEP 1
Rule !
& don't
Approx yes
,
;
STEP 2 : Yp
where
K is constant
,
yn + h
.
ych
+ ky
forget +
diven 4) = 1.
ky
f(x y)
-
Y"
-
,
-
=
t
24
Xy
xz
Yn + 1
=
yn + h
-
=
0
.
Ye
S
1(y()
=
:
Xn + 1
(0 5)(0 + (0)(1))
1 +
.
f(Xn 4)
h
+
=
=
y(1)
=
Ye + (0 3)(Xi + X Y )
,
.
+ 10 3)(0
1
1
.
.
.
5 + 10
.
,
91(1)
5
factoring
↳ perfect
cubic binomial
EX
.
a)"
r3
p
=
*
3r
-
+
3r
Polynomial
Cubic
3ar2 + saar
-
-
2
-
=
=
0
solve for
(r-1)
(multiplicity by 2 (02 =
2)e
*
(r
↑ This is
-
r
-
-
0
=
don't
=
r
forget"to put "C"
y(x)
yo
General solution
Ce
=
3
=
CASE Z
repeated root
EX (0-2) repeated
↳ V
-1
r
,
=
2
=
Ce
,
Yc
=
2
2
,
erration
2)
Y , = Ce2
Yz
2X
=
X
*
,
Yz
XCe
=
Y
-
When r
=
Diff
imp
Step 3) Solve for As
equation
.
and solve for
Step 4) Sub
In
Steps) Y
Clinear >
-
Right Hand Side (MHS)
:
:
.
back to form In
Yn + Yo
RHS
~
Ax + B)
is polynomial equation
ex
.
1" ch
+
=
x+
7
↳
↳
Yp
CASE2
Yp
+
A X
,
+
... +
AmX*)
exponential function
RHS is
:
↳
X"(Ao
=
:
;
I is the multiplicity
ce
**
=
Axe
A KA repated 2 times
.
r
Y
=
, z
=
,
, y Xe
ce
=
-3 4 5
0
=
Y
=
C e"
,
These are
,
,
Yp
CXe
=
, 45 Caxet
:
consider
the same and need to do the
repeated Solution .
* check
carefully if any of these
Y(x) C + (2x
=
+
has the same degree
is
Ce + 2xxe"
+
(,
x
EX
.
,
↳
↳ CASER
-
C e" and ce
↳ CASE3
RHS is
:
Yp
:
=
trigonometric function
X" (Acosex)
multiplication
X
trigonometric
+
or
terms
are
El COsBX + ExsinBX
Bsin(Bx)
addition between Polynomial exponential
,
functions .
↳ Observe the terr In and Yp
If some
:
repeated
,
*
no
term are the same
yp must be
2
2X
Ce"just Add"x" !
atieforcouple oa
0
expo
cubic binomial
,
-
2X
"
Y(x) = Ye + Y
,
Ce
+
:
.
Y , = ce
"
Yp
+
enuation
0
=
yn
=
CASE3 : complex root
=
↳ CASET
charectoristic
(Multiplicity by 3)
0
get
charectoristic
.
-z
:
ger
-
Step 1) Assuming Homogeneous
.
-
L
-
y"
r2ex
=
.
Step 2) Solve for Particular (4p)
Perfect
:
equation
a
↓
r'(r-1)"
vex
Non-Homo linear
↳ Practice
-
'=
you'll
can not be zero
Fans
Higher Order linear Homo
(r
ex
1 > Then
-
y
y =
y
1
=
,
X1
=
.
Co eff
↓
1
Yo + (0 9)(X0 + Xo % )1) Assumin ,
=
Yz
also
=
stop here
to
y(x)
:
.
2) Euler's Method Formula
YA =
-
STEP 3 :
0
=
+
kX
Therefore r2 -r
constant coeff.
X , = Xo + 0 S
solve for
with Constant
↓
-
As
=
y = ce
-
0
Homogenous
Y
let Yp
f(x)
=
-
second order
↓
z
solution form
Second order linear Homo
40 =
0
:
+
↳ Assuming homogeneous
y(0) =
=
K
=
Reverse Product
(YX(
usually this !
Numerical Approx
↓
-
.
e
Equation
↳ equation will become G
this -**
If look like
=
When P(X)
Product Bole
use
Ex
=
5)
1)
f(x 4)
·
like this !
it look
4)
+
Specific linear
form in Step 1
↓
3)
the
like
the equation that looks
Integrating Factor
multiplied by "x"
and