A comprehensive summary of the theorems and applications needed to solve homogeneous, non-homogeneous, systems and partial differential equations with examples and visual aids,
,OLD SEPARABLE DIFFERENTIAL EQUATIONS
sold separable differential equations
separable separated be to find
equations can be
by their variables and
integrated
their solutions .
Example :
① dd- =
22
1-
y2
i. ( 1-
yz) ¥ = x2
i.
11-921 dy = x2 dx
S S
dy
1- x2 dx
: y2 =
g- ¥ =
¥ to
x3
3g y3 t e
-
=
23
3g -
y3
- = C
Solution ! !
② dy Initial conditions :
910) =
-1
DI
=
3221-4×1-2
)
2cg -
l
(
2g 2)
d×
-
= 3×2+4×+2
Zy dx
-2
dy = 3×2+4×+2
S2y -2
dy = 53×2 1-4×1-2 dx
23
y2 2y= t 2×2 1- 2x 1- C
-
x3 1-2×2 1- 2X 1- c
y
=
-2
y
%
-
I =
-
C =3
TZX
i.
YZ -2cg = 23 1-2×2 1- 3
↳ solution ! !
,③
digg Initial condition
ylo)
'
yc
: =
=
11-292
-1292 dy dx
I
ycosx
=
tgtzy dy = Cosx dx
Sty +2g dy
=
Scosxdx
lnyt y2 sinx +
=
c
lnli ) t 12 =
sinloltc
0 1- I = 0 1- C
÷ c. = I
i.
lny tyz
=
sinx 1- 1
CHECKING SOLUTIONS IN DIFF EQNS
① y=3é2✗
'
y t
2g 0 ;
=
2x
i.
be be -250=0
-
+
-
'
. .
0=0 it
②
'
"
Y' =
9g ; y ,
=
e ; yz=é3✗
' '
Yi
=
9e3✗
"= 9 e-
3✗
Yz
i. for qe3✗=9e3✗
y , :
i. True ✓
: for 9é3✗ =
qe
-
3✗
yz
:
-
i. True ✓
yz=xe-2✗
" ' "
③ y t 4y t
4g =o
; y , = e- ,
2e-2✗
'
i.
Yi = -
" -
2x
4e
y , =
for 4e
2✗
-14C Ze
Zx
) 4e
2x
-
i. +
-
y
-
: -
,
=
qe
-
2x -
8e→× + 4E
-
H
=
0
i. True ✓
, yz
'
=
e-
2x -
2xe-2✗
" -4 "
Ze 2C t 4xé2✗
-
=
yz
- -
= -
4e-2✗ + 4xe-2✗
( e- 2x Zxe -2×7
2x
4e -2×1-4 >ce
2x
for yz 1- 4 axe
-
-
i. +
-
-
:
4e_2✗ 4xe-2✗ 1- 4e
2x 2x
4xe
2X
- -
8xe t
-
=
+
-
-
= 0
i. True .
✓
④ xzy
"
t
xy
'
-
y
= dna
① Yi
= x -
lnx
i.
y ,
'
= I -
¥
y
" =
¥2
i. for y, : xz( ¥2) t sell -
¥) -
octlnx
=
I t X -
I -
x + lnx
= lnx
i. True ✓
② Yz = Éc -
lnx
i.
Yz
'
=
-
¥2 -
¥
¥3 ¥2
"
lfz t
=
i. x2( ¥3 + ¥2 ) txt -
¥2 -
¥) -
¥ thnx
2
= I + I -
¥ -
I -
÷ tens
= lnx
i. True ✓
⑤ x2y
" '
✗
y 1-29=0
-
① Yi = rxcosllnx )
y ,
'
=
cosclnxltxc-sinllnxl.sc )
= cos Clnx) -
sinllnx )
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