Consider the wave equation u tt
​
=4(u xx
​
+u yy
​
),(x,y)∈D=[0,3]×[0,2],t>0 with the boundary condition u(x,y,t)=0,t>0,(x,y) on the boundary of D, and the initial conditions: for any (x,y)∈D,
u(x,y,0)=5πsin(3πx)sin(4πy),u t
​
(x,y,0)=40πsin(3πx)sin(4πy)
​
The solution is given by u(x,y,t)=∑ n=1
[infinity]
​
∑ m=1
[infinity]
​
[A mn
​
cos(λ mn
​
t)+B man
​
sin(λ mn
​
t)]sin 3
mπx
​
sin 2
nπy
​
, where λ mn
​
=2π 9
m 2
​
+ 4
n 2
​
​
Find the coefficient B 9,8.
​
a) 3 b) 5 c) 2 d) 4 e) 6 The electrostatic potential u(r) (in volts) between two coaxial cylinders of radii r 1
​
=e and r 2
​
=e 5
satisfies the equatiol u rr
​
+ r
1
​
u r
​
=0 The potentials carried by the cylinders are u(e)=7 and u(e 5
)=15, respectively. Find the electrostatic potential u(e 4
) a) 9 b) 11 c) 14 d) 10 e) 13 Consider the differential equation: x 2
(x+1)y ′′
+5x(x+1)y ′
−6y=0,x>0, near x 0
​
=0. Let r 1
​
,r 2
​
be the two roots of the indicial equation, then r 1
​
+r 2
​
= a) −4 b) −5 c) −3 d) −2 e) −6