
1016 HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
11.5.5 Equation of the Form
∂
4
w
∂x
4
+
∂
4
w
∂y
4
+ kw = Φ(x, y)
◮ Particular solutions of the homogeneous equation (Φ ≡ 0):
w(x, y) =
A sin(λx) + B cos(λx) + C sinh(λx) + D cosh(λx)
exp(βy) sin(βy),
w(x, y) =
A sin(λx) + B cos(λx) + C sinh(λx) + D cosh(λx)
exp(βy) cos(βy),
w(x, y) =
A sin(λx) + B cos(λx) + C sinh(λx) + D cosh(λx)
exp(−βy) sin(βy),
w(x, y) =
A sin(λx) + B cos(λx) + C sinh(λx) + D cosh(λx)
exp(−βy) cos(βy),
where β =
1
√
2
(λ
4
+ k)
1/4
; A, B, C, D, and λ are arbitrary constants.
◮ Domain: 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
. Boundary value problems.
1
◦
. Consider problems in the rectangular domain with various homogeneo ...