1. Solve by the method of undetermined coefficients and by the method of variation of parameters. (4)
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1 7 Differeial equaios Review Solve by he mehod of udeermied coefficies ad by he mehod of variaio of parameers (4) y y = si Soluio; we firs solve he homogeeous equaio (4) y y = 4 The correspodig characerisic equaio λ = has soluios,, i, iwhece he geeral soluio of he homogeeous equaio is y = Ce + Ce + C cos+ C si 3 4 To fid a paricular soluio of our o-homogeeous equaio we ca use eiher he mehod of udeermied coefficies discussed i Secio 43 or he mehod of variaio of parameers from Secio 44 Le us cosider boh ways (a) Mehod of udeermied coefficies Because cos ad si are soluios of homogeeous equaio ad he righ par is si we will ry o fid a paricular soluio of he formy( ) = A ( cos+ B si ) I is easy o compue ha Y (4) ( ) = ( Acos+ Bsi ) + 4Asi 4Bcoswhece Y (4) () Y() = 4Asi 4Bcosad we ca ake A=, B=, Y( ) = cos 4 4 The geeral soluio is y = Ce + Ce + C3cos+ C4si + cos 4 (b) Mehod of variaio of parameers We will use formula () o page 39 We sar wih compuig he Wroskia (see formula (7) o page ) e e cos si e e si cos W () = e e cos si e e si cos By Abel s formula (see Problem o page 3) W ( ) is a cosa fucio of I paricular we ca compue W () W () W() = = = = 8 Durig he compuaio above o he firs sep we subraced he firs row from he oher rows ad o he secod sep we subraced he secod row from he las oe These operaios do o chage he deermia
2 Nex we have o compue he deermias W (), W (), W (), ad W () where 3 4 Wm ( ), m=,,3,4is obaied from W () by subsiuig colum umber m by he colum So we have e cos si e cos si e si cos W () = = e si cos = e cos si e cos si e si cos cos si cos si si cos si cos (cos si ) e = e = e = e + = cos si e cos si e cos si e si cos W () = = e si cos = e cos si e cos si e si cos e cos si = e si cos = e e e e si e e si e e cos W3 () = = e e cos = e e si e e si e e cos e e si = e e cos = 4si si
3 e e cos e e cos e e si W4 () = = e e si = e e cos e e cos e e si e e cos = e e si = 4cos cos Applyig formula () o page 39 we ge s s Y ( ) = e si s( e ) ds e si s( e ) ds cos si s( 4si s) ds si si s(4cos s) ds Usig he formulas s si s e ds = e (cos+ si ), si cos si si, si cos si cos, s si s e ds= e (cos si ), sds = = 4 s sds= = 4 4 we obai Y ( ) = e (cos+ si ) e+ + e(cos si ) e cos si cos si + si cos Takig io cosideraio ha si cos cos si = si( ) = si we ge Y () = e e cos si+ cos Because he firs four erms i he expressio above are soluios of he homogeeous equaio we ca ake a paricular soluio i he form Y () = cos 4 i complee agreeme wih he soluio we go i par (a)
4 Solve i series ear o poi Fid he radius of covergece of your soluio ( x ) y + xy y = Soluio; poi is a ordiary poi for our equaio because P () = Therefore we look for soluios aalyic i some eighborhood of, ie soluios of he form The y( x) = a x = ( ) =, ( ) = ( ), = = y x ax y x ax ad pluggig hese expressios io our equaio we ge or ( ) a( x x ) + ax ax = = = = ( ) a ( + )( + ) a + x = = Coefficies by ay power of x i he lef par should be equal o, whece ( ) a ( + )( + ) a =, =,,K From he above relaios we easily ge ha for odd values of he coefficie is a arbirary umber ad a3 = a5 = a7 = K = For eve values of we have he recurre relaio a+ = a, =,,4, K + Therefore 3 K ( 3) 3 K ( 3) a = a ad a = a = a =,3, 4, K 4 K! The geeral soluio of our equaio is he liear combiaio of wo liearly idepede soluios 3 ( 3) y = ax + a x x K =! To fid he radius of covergece of he series i he las expressio we use he raio es; + a+ x lim = lim x = x a x + whece he series coverges if x < ad is radius of covergece is + a
5 3 Solve usig he Laplace Trasform y + y = f(), y() =, y () =,, < π where f( ) =, π < π, π Soluio; applyig formula (3) o page 35 we see ha he Laplace rasform of y is L( y )() s = s L ( y)() s whece ( s + ) L( y)( s) = L ( f)( s ) To fid he Laplace rasform of f we jus use he defiiio (see also page 36) π s π π s π s L s s e e e ( f)( s) = e f( ) d = e d = = s s π Now we see ha πs πs e e L ( y)( s) = + ss ( + ) s + Therefore πs πs - e e y() = L + () ss ( + ) s + To compue he righ par of he previous expressio we firs have o wrie as he ss ( + ) sum of parial fracios A Bs+ C = + ss ( + ) s s + To fid he coefficies A, B, ad C we wrie = As ( + ) + Bs + Cs Pluggig i s = we ge A = ; ad pluggig i A= iwe ge B + Ci = whece B = ad C = Hece, s = ss ( + ) s s +, ad - s s s s s s y() e π - () e π - () e π - π () e - = L L + () + () L L L s s + s s + s + - From formula 5 i Table 6 o page 39 we ge L () si s = Combiig + formulas, 6, ad 3 from he same able we obai he expressios π
6 - π s s L e () ()cos( ) ()cos, = uπ s π = uπ s s + - π s s L e () u = π()cos( s π ) = uπ()cos s, s + - π s L e () = uπ (), ad s - π s L e () = uπ () s, < c, Recall ha (see formula () o page 35) uc () = c Fially we ge he, c, soluio of our equaio Or y() = u () u () + u ()cos + u ()cos + si π π π π si, < π, y () = + cos+ si, π < π, cos+ si, π
7 4 Solve i series ear o poi xy xy + x+ y= ( ) Soluio; we apply Theorem 57 o page 9 I oaios acceped i his heorem we have xp ( x ) = whece p =, p = p = p = K = ; ex we 3 4 x + have xqx ( ) = whece q = q =, q = q 3 = q 4 = K = The idicial equaio is 3 Fr () = rr ( ) + pr + q = r r+ = The soluios of his quadraic equaio are r =, r = Applyig formula () o page 9we ge whece ad y ( x) x a x = + = = + y ( x) = + ( + ) a x = y ( x) = ( + ) a x Pluggig i hese expressios io he origial equaio ad combiig like erms we ge 3 = 3 3 ax (a ) x [ ( ) a a ] x = From here we easily obai ha a = a3 = a5 = K = ad a = ( ) = ( ) K ()59 K (4+ ) (!)59 K (4+ ) Now le us use formula () o page 9 The / + / ( ) = +, = y x x b x x ad y x x b x x / / ( ) = + ( + /), > =
8 y x x b x x 3/ 3/ ( ) = + ( /4), > 4 = Agai, pluggig i hese expressios io he origial equaio we ge Therefore b = b3 = b5 = K = ad ( ) b =!3 7 K ( ) Fially we have + y( x) = A x+ ( ) x ( ) + B x + x =!5 9 K (+ ) =!3 7 K ( ) /
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