The C.F. for the PDE (D2 – 4D'2)z = ? _ x?
X
y
C.F.= 0,(y+ 2x) + O2(y + 2x)
Answers
Step-by-step explanation:
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Answer:
162
The most general form of a linear partial differential equation
of order n is
(
A0
∂
n
z
∂ x
n
+ A1
∂
n
z
∂ x
n−1
∂ y
+A2
∂
n
z
∂ x
n−2
∂
2
y
+...+ An
∂
n
z
∂ y
n )
+(
B0
∂
n−1
z
∂ x
n−1
+B1
∂
n−1
z
∂ x
n−2
∂ y
+…+Bn−1
∂
n−1
z
∂ y
n−1 )
+...+(M
∂ z
∂ x
+N
∂ z
∂ y )
+Pz=F(x , y ) …(1)
where the coefficients A0,A1,A2…,An, B0, B1,…, Bn–1,…, M, N, P
are either constants or functions of x and y.
When the coefficients of various terms namely A0, A1, A2
…,An,B0,B1,…,Bn–1,…,M,N and P in the above partial differential
equation are all constants, then equation (1) is known as linear
partial differential equation with constant coefficients.
For convenience, if we use the symbolic notations D for ∂
∂ x
and D ' for ∂
∂ y
, then (1) can be re-written as
[( A0 D
n
+ A1D
n−1 D
'
+ A2D
n−2 D'
2
+...+ An D '
n
)
+(B0D
n−1
+B1D
n−2 D
'
+...+Bn−1 D'
n−1
)
+...+( MD+ND' )+P] z=F ( x , y ) …(2)
or briefly f ( D , D
'
) z=F(x , y) …(3)
where f (D ,D
'
) denotes a partial differential operator of the type