CHAPTER I
CALCULUS
D- OPERATORS
,Linear Equations
farm
play qct )
'
→
y
=
: +
solution "fµQdt
'
'
→
µ +
CM
=
y
:
where µ is the INTEGRATING FACTOR
and is equal to qfpatdt
Solution
§
can
be separated Y
=
Yi t
Yo
he b
fyadt
'
µ
'
'
cy
'
→
produces a zero
→ particular solution solution
→ general solution
→ known as homogeneous
mm
Relation to Matrices :
→
The matrix equation AX =
B can also be split
X =
X ,
+ Xo
pahtcular ↳
arbitrary
solution constants
¥F This similarity lies in the concept of a LINEAR OPERATOR
, Linear Operators
input →m
Fatsf. output
] -
.
A linear operator satisfies 2 conditions :
P ( cy ) cpcy ) 2
Pkgitdya)
1
=
2 Patty . ) =
My , ,
+
pay ,
f =(Py ,
t
dpyz
linear
eg
.
Prove Pc g) =
DYE +
zycatlts is a
operator .
!1!
pay ) =
dattky ) + 2C cykotlt ) *
by rules of
=
cadet + czycatlt ) differentiation
=
c[ dot
+
Zycotlts ]
= CPC y )
!2! P( y ,
+
ya )
=
It ( yityz ) Zlyityz ) ( at Ct )
cattt ) Zyzcotlt )
# datyzt
+
Yi 2y
=
+ ,
=
Yi
⇐
+2g ,
catt
] + [ datyatzyeott ]
linear
= ply ) , + Plys) . ; operator is
CALCULUS
D- OPERATORS
,Linear Equations
farm
play qct )
'
→
y
=
: +
solution "fµQdt
'
'
→
µ +
CM
=
y
:
where µ is the INTEGRATING FACTOR
and is equal to qfpatdt
Solution
§
can
be separated Y
=
Yi t
Yo
he b
fyadt
'
µ
'
'
cy
'
→
produces a zero
→ particular solution solution
→ general solution
→ known as homogeneous
mm
Relation to Matrices :
→
The matrix equation AX =
B can also be split
X =
X ,
+ Xo
pahtcular ↳
arbitrary
solution constants
¥F This similarity lies in the concept of a LINEAR OPERATOR
, Linear Operators
input →m
Fatsf. output
] -
.
A linear operator satisfies 2 conditions :
P ( cy ) cpcy ) 2
Pkgitdya)
1
=
2 Patty . ) =
My , ,
+
pay ,
f =(Py ,
t
dpyz
linear
eg
.
Prove Pc g) =
DYE +
zycatlts is a
operator .
!1!
pay ) =
dattky ) + 2C cykotlt ) *
by rules of
=
cadet + czycatlt ) differentiation
=
c[ dot
+
Zycotlts ]
= CPC y )
!2! P( y ,
+
ya )
=
It ( yityz ) Zlyityz ) ( at Ct )
cattt ) Zyzcotlt )
# datyzt
+
Yi 2y
=
+ ,
=
Yi
⇐
+2g ,
catt
] + [ datyatzyeott ]
linear
= ply ) , + Plys) . ; operator is
