Calculus Chapter 1

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