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1 Math 418, Fall 2018, Midterm Prf. Sherman Name nstructins: This is a 50 minute exam. Fr credit yu must shw all relevant wrk n each prblem. Yu may nt use any calculatrs, phnes, bks, ntes, etc. There are a ttal f 60 pints available (12 n each page). f yu need extra space please use the backs f the pages and clearly indicate that yur wrk cntinues there. Prblem 1. Cnsider the eigenvalue prblem X 00 + X 0 (0) + X(0) = 0 X 0 (1) + 2X(1) = 0 X =0, 0 <x<1, (a) (2 pints) Shw directly that = 0 is nt an eigenvalue. Ya x a X Axe B X'l AtB T X At AtB A Ba (b) (10 pints) Find, frm first principles, the negative eigenvalues graphically. Yu shuld set = 2 fr >0, then derive an equatin f the frm tanh( ) = g( ). Plt bth sides f the equatin n the same graph, clearly pltting the relevant asympttes, and indicate any intersectin pints. Yu will need t cnsider the slpes f bth tanh( ) and g( ) at = 0 (Hint: d d tanh( ) =0 = 1) X g g 8 2 s Zs yr xfyy.sny i z2schhyescsh's 2C scshyesukh 8 Lars 8 28 tftaahsy.c tair E gyef gsh E si Eire

2 (c) (12 pints) Find, frm first principles, the psitive eigenvalues graphically. Yu shuld set = 2 fr >0, then derive an equatin f the frm tan( )=f( ). Plt bth sides f the equatin n the same graph, fr 0 apple apple 3, clearly label all relevant asympttes, and indicate any intersectin pints. Yu will need t cnsider the slpes f bth tan( ) and f( ) at d = 0 (Hint: d tan( ) =0 = 1). Label the intersectin pints 1 < 2 < 3 <. Find an interval fr each n, and find an apprximate value fr n when n is large. 1 p2 X p'x X'cstXcs Xifcspx Bsihpx p B t A A PD ye BL pcspxtschpx X'a e2xcd p's.lu 3i pcspt2 pcsp isckp 0 p'taupep tarp tamp P 4 flp p f'c z s 1 is ti ii and p z ult fr a large

3 Prblem 2. Let 8 1 if 0 <x< /2 '(x) = 0 if /2 <x< (a) (6 pints) Find the Furier-sine series f '(x) n the interval [0, ]. Find the value f the nth ce ut the first fur nnzer terms f the series. F S N E Bushhfx n Ba na l 7 l a cient, and als write iii Yt C csnxlth Y s E l cs c i ci cs E s (b) (3 pints) Sketch a graph f the functin F (x) that the Furier-sine series cnverges t fr each 0 apple x apple. 0 tan (c) (3 pints) Parseval s equality can be used t find the value f a certain infinite sum. Find that series and its sum.

4 Prblem 3. (12 pints) Let u satisfy the fllwing inhmgeneus prblem: u t u xx =0, 0 <x<, 0 <t u(0,t)=0, 0 <t u(, t) =t/(1 + t 2 ), 0 <t u(x, 0) = a n sin(nx), 0 <x< where the a n s are knwn given ce cients. f we immediately expand a slutin in a Furier-sine series as u(x, t) = u n (t)sin(nx), then find an ODE and an initial cnditin satisfied by each u n (t) fr n =1, 2, 3,... ate Eaicasmux 7Lik p U wuct Schux where i wnch jefjux.is uxdx e STU sihnxdxtfu.snnx u.ucsnx j Ef n'siu shuxdxtfz.tn i i5 u Unct E T t2 PDB ce sle ex Eun sinux nt uicts n'units t sinux ODE UCL7 iuiuu.cat f nut C UnC7

5 Prblem 4. (2 pints fr each part belw). Cnsider an eigenvalue prblem X 00 + X =0, a < x < b with symmetric bundary cnditins at x = a, b. Give shrt answer respnses t each: (a) True r False: All the eigenvalues must be real. True (b) True r False: All the eigenvalues must be nnnegative. False fr example see Rbin BC earlies in this exam (c) f 1 and 2 are tw di erent eigenvalues, with crrespnding eigenfunctins X 1,X 2, what can we say abut the functins X 1,X 2? they are rthgnal JbaX Xzd (d) Suppse '(x) is a functin defined n [a, b] such that R b a '(x)2 dx is finite. Assume X 1,X 2,X 3,... frm a cmplete rthgnal set f eigenfunctins. f we wish t expand ' in a Furier series P A n X n (x) f the eigenfunctins, what is the frmula fr the ce cients A n? Au t Y wheuelf g7 f fcxigcxs (e) n the ntatin f the previus part, True r False: fr each a apple x apple b we must have '(x) = A n X n (x). False fun example cnsider a classical F S (f) n the ntatin f the previus part, True r False: at a pet where e has a jump disc Z b ('(x) a A n X n (x)) 2 dx = 0. True