INTRa. TO DIFFERENTIAL EQUATIONS, Spring 2009, Test 2A

Section Name -""X>""-~:....:\.I:...:.i\i.!..:. (;::..:."'"--=.l~ _

Instructions. You are allowed to use one 8 1/2 x 11 inch sheet of paper of notes. No books orelectronic equipment (including calculators, PDAs, computers, MP3 players, or cell phones)are allowed. Do not collaborate in any way. In order to receive credit, your answers must beclear, legible, and coherent.

Problems 1-3 deal with the two-point boundary value problem

X" + AX = 0, X'(O) = 0, X'(7r) = °C 1. When A = 0, the nonzero solutions of (0) are

A) X(x) = Ax + B B) X(x) exx C) X(x) ex1E) The A = ° case has no nonzero solutions.

(0)

D) X (x) = A cosnx + B sin nx

F 2. When A < 0, the problem (0) has nonzero solutionsA) only when A = -n2, n = 1,2, ; in that case X(x) = Aenx + Be-nx

B) only when A = -n2, n = 1,2, ; in that case X(x) excosnxC) only when A = -n2, n = 1,2, ; in that case X(x) exsin nxD) only when A = - (n + ~)2 ,n = 0,1,2, ; in that case X(x) excosnx

E) only when A = - (n + ~)2 ,n = 0,1,2, ; in that case X(x) exsin nxF) The A < ° case has no nonzero solutions.

XIx):;. A eM)( '\ () .e-A.-r)( or ~ (." co!, \...M Jc' + C"Z. S,,,- '" A.4k

6 3. For the case A > 0, the problem (0) has nonzero solutionsA) only when A = n2,n = 1,2, ; in that case X(x) = Aenx + Be-nx

B) only when A = n2,n = 1,2, ; in that case X(x) excosnxC) only when A = n2,n = 1,2, ; in that case X(x) exsinnxD) only when A = (n + ~)2 ,n = 0,1,2, ; in that case X(x) excosnxE) only when A = (n + ~)2 ,n = 0,1,2, ; in that case X(x) exsinnxF) The A > ° case has no nonzero solutions.

X ()r)= A c 0;) ~>< 'T v3 ~'".. '('1)0'

'1/(~):: -:-A A..t &" ''''' M)r +- '3 --'t ('o~ A,,"

o = XI fo) -:; 6.-4 '"="') X3 ~ 0

- X '() A .-M'" -=- ",,'IT -:;::..")M6 - 'IT -= .- . A..t S'V\ ~ '1r -=:) .

~1.+~)..=o

G 4. In solving the two-point boundary value problem X" + 2X' + AX = 0,X (0) = °= X (L ), the cases that must be considered areA) A = 0, A < 0, A > ° B) A = 1, A < 1, A > 1 C) A = 2, A < 2, A > 2D) A = 4, A < 4, A> 4 E) All values of A can be considered simultaneously.

(' '1 .+- 2 (".~~ =0 _) ., -:.- 2. i J 4 - \f), -:: -I t rT=>:2.

Problems 5 - 7 concern the problem

Uxx + Uyy = 0, U(X,O) = f(x), u(x,7r) = g(x), u(O, y) = 0 = u(7r,y), (EB)00

whose solution is of the form u(x, y) = L (Aneny + Bne-ny) sin nx.n=l

~5. How should f and g be extended outside the interval [0,7r]?A) f and g should be extended as odd periodic functions with period 7r.B) f and g should be extended as odd periodic functions with period 27r.C) f and g should be extended as even periodic functions with period 7r.D) f and g should be extended as even periodic functions with period 27r.E) f should be extended as an odd periodic function and g should be extended as an evenperiodic funciton.F) f should be extended as an even periodic function and f should be extended as an oddperiodic function.

___ ·6. For f(x) = sin2x and g(x) = 3sin4x, what is true about the As and Bs?A) A2 = 1, A4 = 3, and all other As and Bs are zero.B) B2 = 1, B4 = 3, and all other As and Bs are zero.

C) They are obtained by solving {1~!~~~and all other As and Bs are zero.

D) They are obtained by solving {AA2 !7rB2 B 1_47r _ 3 and all other As and Bs are zero.4e + 4e -

) . . { A2 + B2 = 1 { A4 +B4 = 0E They are obtamed by solvmg A 27r B 27r _ 0 and A 47r B -47r _ 3 and all2e + 2e - 4e + 4e -

other As and B s are zero. _ ""S /, ) /. Y)'1j'" -,.."if) ,Si ....~)(::: L.. t:A",+~ s~V\rlY 3~'"'l4)c:; :2c.....AJlt!. .f (3"e ~"".,x

~ 7. If we find that the solution to problem (EB)has B2 = 2, A3 = 3, B4 = 4, and allother As and Bs are zero, then the solution to problem (E9) isA) 2sin2x + 3sin3x + 4sin4x B) 2e-2y sin2x + 3e3y sin3x + 4e-4y sin4xC) 2 (e2Y + e-2y) sin2x + 3 (e3Y + e-3y) sin3x + 4 (e4Y + e-4y) sin4xD) The solution cannot be determined from this information.

(10 points) 8. Sketch an approximation to the function f. Label important points on youraxes, and extend your drawing for several periods.

1000 1/2 1

f(x) = ~o + Lancosn7rx where an = -21 xcos(n7rx)dx+j cos(n7rx)dx.n=l '\ 0 { 1/2

L -=. I ~ tl.",::: 2. ( Ii (><) CO! '1\ Trx' J)c -=-:> 2t;'{~):::: - Qx Cv" [OJ 112JJo . [II "A I2-J 'J

... -:1'lzJ,. t. J ~ ~ ,~/x) ':. (~~, ,,~[PJ Y2JlY b" [V2 I]2 I

C"'""c\..\i:\t,, <) L2-) P<r\OJ.,c.~) eve.",

Problems 9 - 11 deal with Ut = XUxx, ux(O, t) = 0 = u(3, t), U(x, 0) = 1. (#)

_-=C=--9. The two-point boundary value problem associated with (#) involves solving theordinary differential equationA) X" + AX = 0 B) X" + AXX = 0 C) xX" + AX = 0 D) xX" + A = 0E) The variables cannot be separated for this problem.

1~li~x4/'T -=::> T':: xX,,11 -= -). _) x XI' +~:K =0

XT T X"~1O. The boundary conditions for the two-point boundary value problem in 9) are

A) X(O) = 0 = X(3) B) X(O) = 0 = X'(3) C) X'(O) = 0 = X(3)D) X'(O) = 0, X(O) = 1 E) X'(O) = 0 = X(3), X(O) = 1

o -::ux,/oJ t) ':: ,Xl/Q) Tll) 0 -:: ti (3,t) ::"K I3)'1 lof)

~11. The other ordinary differential equation that arises in solving (#) isA) T' + AT = 0 B) T' - AT = 0 C) T' - A = 0 D) T' + A = 0E) The variables still cannot be separated for this problem.

o 12. If u(x, t) = X(x)T(t) satisfies the differential equation and the homogeneous bound-ary conditions in the problem

Utt + Ut = Uxx, u(O, t) = 0 = u(7f, t), U(x, 0) = f(x), Ut(x, 0) = g(x),

the equation for T isA) Til +T' = 0 B) Til +T' = _n2

X'T" ,t- XT' :;;X/I'T

C) Til +n2T' +T = 0 D) Til +T' +n2T = 0,

-;>TifT I Xl/ _-+-:;;.- ~)..T T X

IfX .~~ )( :::0Xfo\ ;::0 ~Xtrr)

,-" II ·t-T +~'\:::.o "''''> '<!: f()t ",,,- i..~)~ = n'"

o 13. Heat conduction in a circular ring can be modeled by the problem

Ut = Uxx, u( -7r, t) = u(7r, t), U(x, 0) = f(x).

The two-point boundary value problem for problem (**) has periodic boundary conditions:X" + AX = 0, X( -7f) = X(7r). This two-point boundary value problem has nonzero so-lutions only when A = n2, n = 0,1,2, ... , and the corresponding solutions X are Xn(x) =acosnx+bsinnx (for n = 0, Xo =const.). Moreover, the equation T' +n2T = 0 has solutionsTn(t) exe-n2t. Solutions to (**) are then of the form

00 00

A) Lbne-n2tsinnx B) Lane-n2tcosnxn=l n=l

00

ao """ 2tC) 2 + 6 ane-n cosnxn=l

00

E) ao +bot+ Le-n2t(ancosnx+ bnsinnx)n=l

B 14. Which answer best describes the correct approach for solving the followingproblem?

Uxx + Uyy = 0, u(x,O) = 0, u(x, 'if) = 1, u(O, y) = 0, u(7r, y) = 2,

A) Solve the two-point boundary value problem X" + AX = 0, X(O) = 0, X(7r) = 1, thenthe two-point boundary value problem Y" - AY = 0, YeO) = 0, Y(7r) = 2; then the solutionto (*) is u(x, y) = 2:::nCnXn(x)Yn(y).B) First solve the problem Vxx + Vyy = 0, v(x,O) = 0, v(x,7r) = 1, v(O, y) = 0, v(7r, y) = 0;then solve Wxx +Wyy = 0, w(x,O) = 0, w(x,7r) = 0, w(O,y) = 0, w('if,y) = 2; then thesolution to (*) is u(x,y) = v(x,y) +w(x,y).C) A) and B) are equivalent and either approach can be used to solve problem (*).D) Neither A) nor B) is correct; problem (*) cannot be solved by the method of separationof variables,

I082 ~

o o

Problems 15 and 16 deal with the problem

Ut = Uxx, u(O, t) = 1, u(7r, t) = 1, U(x, 0) = 0

L15. For large t, the solution u(x, t) to the problem (EB)

(EB)

A) approaches a nonzero constant, B) has the form of a traveling wave, C) goes to zero,D) goes to infinity, E) cannot be determined from the information given.

Lt/x,i-) ~ ,,(,..)+ "-.J{Xt-l) V>;,,><::: 0

1o £r \a.r.,).....-t::

~16. Which of the following is NOT involved in solving problem (EB)?A) trying a solution of the form X(x)T(t)B) solving the two-point boundary value problem X" + AX = 0, X(O) = 1, X(7r) = 1C) solving T' + n2T = 0D) evaluating the expression 2:::cnXn(x)Tn(t) at t = 0