HOMEWORK 2 SOLUTIONS
Section 1.5 : #1, 2, 3, 4(a) - (b) #1) Solve the boundary value problem: uxx + u = 0 u(0) = 0 and u(L) = 0. Answer: The solution to the ODE is: u(x) = A sin(Bx + C) If A = 0, then the solution is u(x) 0. If A = 0, then we can s
HOMEWORK 1 SOLUTIONS
Section 1.1 : #2, 3, 11, 12 #2) Which of the following operators are linear? (This is accomplished by checking if Lu = cLu and L(u + v) = Lu + LV are satisfied). (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux
Solve au x + bu y = f ( x, y ) were f is a given function. Write the solution in the form
u ( x, y ) = a 2 + b 2
(
) " fds + g (bx ! ay ) .
!1/2 L
Using the coordinate method:
u x = au x ! + bu y ! x ! = ax + by ! y! = bx " ay u y = bu x ! " au y
n !1 % " utt = c 2 $ urr + ur ' # & r
u ( r,t ) = ! ( r ) f ( t " # ( r ) ut = ! ( r ) f $ ( t " # ( r ) ) utt = ! ( r ) f $ ( t " # ( r )
ur = ! $ ( r ) f ( t " # ( r ) " # $ ( r ) ! ( r ) f $ ( t " # ( r ) urr = ! $ ( r ) f ( t " # ( r ) " 2 # $ (