Question

the complementary function of (D-D'-1) (D-D'-2)Z=e^2x- y is

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

Consider the following homogeneous linear partial

differential equation of order n with constant coefficients:

( A0D

n

+ A1D

n−1 D

'

+...+ AnD '

n

) z=F (x , y )

i.e. f ( D , D

'

) z=F( x , y ) …(1)

where the coefficients A0

, A1

,…., An are all constants.

The complementary function (C.F.) of (1) is the general

solution of f ( D , D' )=0 i.e.,

( A0D

n

+ A1D

n−1 D

'

+...+ AnD '

n

) z=0 …(2)

which can also be factorized in the following form:

[( D−m1D ') ( D−m2D ') …( D−mn D ') ]z=0 …(3)

where m1

,m2

,….,mn are some constants.

Clearly, the solution of any one of the differential equations

( D−m1D ') z=0,( D−m2 D' ) z=0,…,( D−mn D' ) z=0 …(4)

is also a solution of equation (3).

Now, let us find the general solution of the linear partial

differential equation of the type ( D−mD') z=0 as follows:

The equation ( D−mD') z=0 can be re-written as

∂ z

∂ x

−m

∂ z

∂ y

=0 or p−mq=0 …(5)

which is in Lagrange’s form Pp+Qq=R.

∴ Lagrange’s auxiliary equations for (5) are